%------------------------------------------------------------------------------
% File     : ITP280_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Space 00398_018442
% 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    : 0076_VEBT_Space_00398_018442 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11395 (2607 unt;1519 typ;   0 def)
%            Number of atoms       : 28179 (8026 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 20351 (2048   ~; 329   |;2340   &)
%                                         (2043 <=>;13591  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   31 (   2 avg)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  : 1254 (1074   >; 180   *;   0   +;   0  <<)
%            Number of predicates  :  228 ( 225 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1283 (1283 usr;  73 con; 0-8 aty)
%            Number of variables   : 30978 (27986   !; 659   ?;30978   :)
%                                         (2333  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 17:10:56.376
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

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

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

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

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

tff(ty_t_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 (1501)
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_Obot,type,
    bot: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

tff(sy_cl_Enum_Ofinite__distrib__lattice,type,
    finite8700451911770168679attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot3__space,type,
    topological_t3_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ot4__space,type,
    topological_t4_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
    counta4013691401010221786attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__nu____,type,
    aTP_Lamp_nu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__og____,type,
    aTP_Lamp_og: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oh____,type,
    aTP_Lamp_oh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oi____,type,
    aTP_Lamp_oi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__pj____,type,
    aTP_Lamp_pj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pk____,type,
    aTP_Lamp_pk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pl____,type,
    aTP_Lamp_pl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pm____,type,
    aTP_Lamp_pm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : 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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
    bNF_We4925052301507509544nc_map: 
      !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,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_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 ) > fun(int,int) ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,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_Oand__num,type,
    bit_un1837492267222099188nd_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
    bit_un5425074673868309765um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
    bit_un3595099601533988841um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),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_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_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

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

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

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

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

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

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

tff(sy_c_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_Countable__Set_Ocountable,type,
    countable_countable: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Countable__Set_Ofrom__nat__into,type,
    counta4804993851260445106t_into: 
      !>[A: $tType] : ( set(A) > fun(nat,A) ) ).

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

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

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

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,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_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

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

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

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

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_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_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_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_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_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),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] : ( set(A) > A ) ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

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

tff(sy_c_Groups__List_Omonoid__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_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

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

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

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

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

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

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),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_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,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_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

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

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

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,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_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

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

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

tff(sy_c_List_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_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,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_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_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > 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__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

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

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

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

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

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

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,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__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(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_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__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_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > 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_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_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_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__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__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),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_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun(num,num) ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

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

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : ( A > option(A) ) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

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

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

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

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

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

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,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_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

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

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_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_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

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_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_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__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
    real_V4916620083959148203axioms: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_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_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 ) > fun(A,real) ) ).

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

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

tff(sy_c_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_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) > fun(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_Oremove,type,
    remove: 
      !>[A: $tType] : ( ( A * set(A) ) > set(A) ) ).

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(A) ) > filter(A) ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oclosed,type,
    topolo7761053866217962861closed: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
    topolo2193935891317330818ompact: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
    topolo1966860045006549960nected: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
    topolo6863149650580417670ergent: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
    topolo2479028161051973599mplete: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
    topolo7806501430040627800ormity: 
      !>[A: $tType] : filter(product_prod(A,A)) ).

tff(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
    topolo6026614971017936543ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_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_Orec__VEBT,type,
    vEBT_rec_VEBT: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))) * fun(bool,fun(bool,A)) * vEBT_VEBT ) > A ) ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun(vEBT_VEBT,fun(nat,bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).

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_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_Oinit__seg__of,type,
    init_seg_of: 
      !>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).

tff(sy_c_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_va____,type,
    va: nat ).

% Relevant facts (9084)
tff(fact_0_False,axiom,
    ~ 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)))) ).

% False
tff(fact_1__C3_OIH_C_I3_J,axiom,
    ! [X: 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,va))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,X)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(aa(nat,nat,suc,X)))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ) ) ).

% "3.IH"(3)
tff(fact_2__C3_OIH_C_I4_J,axiom,
    ! [X: 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,va))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(X))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ) ) ).

% "3.IH"(4)
tff(fact_3__C3_OIH_C_I1_J,axiom,
    ! [X: 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,va))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(X))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ) ) ).

% "3.IH"(1)
tff(fact_4_space__cnt,axiom,
    ! [T2: vEBT_VEBT] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)))) ).

% space_cnt
tff(fact_5_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_6_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: 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))),A2))
        <=> 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),A2),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_7_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_8_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_9_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_10_semiring__norm_I84_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit1,N) ).

% semiring_norm(84)
tff(fact_11_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit1,M) != aa(num,num,bit0,N) ).

% semiring_norm(89)
tff(fact_12_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit0,M) != aa(num,num,bit1,N) ).

% semiring_norm(88)
tff(fact_13_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_14_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_15_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% of_nat_le_iff
tff(fact_16_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M = N ) ) ) ).

% numeral_eq_iff
tff(fact_17_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ).

% semiring_norm(13)
tff(fact_18_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(71)
tff(fact_19_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit0,M) = aa(num,num,bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_20_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_21_semiring__norm_I12_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).

% semiring_norm(12)
tff(fact_22_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),one2),N)) ).

% semiring_norm(68)
tff(fact_23_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_24_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_25_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_26_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(73)
tff(fact_27_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_28_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% numeral_le_iff
tff(fact_29_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_30_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% numeral_times_numeral
tff(fact_31_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_32_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),one2)) ).

% semiring_norm(69)
tff(fact_33_semiring__norm_I83_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit0,N) ).

% semiring_norm(83)
tff(fact_34_semiring__norm_I85_J,axiom,
    ! [M: num] : aa(num,num,bit0,M) != one2 ).

% semiring_norm(85)
tff(fact_35_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,N))) ).

% semiring_norm(14)
tff(fact_36_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_37_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(72)
tff(fact_38_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),one2)) ).

% semiring_norm(70)
tff(fact_39_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% Suc_le_mono
tff(fact_40_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).

% le_cube
tff(fact_41_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_42_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).

% le_trans
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),collect(A,P)))
    <=> pp(aa(A,bool,P,A2)) ) ).

% mem_Collect_eq
tff(fact_44_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A3)) = A3 ).

% Collect_mem_eq
tff(fact_45_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_46_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F2,X3) = aa(A,B,G,X3)
     => ( F2 = G ) ) ).

% ext
tff(fact_47_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% eq_imp_le
tff(fact_48_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_49_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_50_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_51_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( 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),J),L))) ) ) ).

% mult_le_mono
tff(fact_52_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => 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),J),K))) ) ).

% mult_le_mono1
tff(fact_53_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => 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),J))) ) ).

% mult_le_mono2
tff(fact_54_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% nat_le_linear
tff(fact_55_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: 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)),M)),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),M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_56_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y: nat] :
            ( pp(aa(nat,bool,P,Y))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
            & ! [Y3: nat] :
                ( pp(aa(nat,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_57_div__times__less__eq__dividend,axiom,
    ! [M: 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),M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_58_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: 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),M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_59_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_60_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_61_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),X3))
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Y))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,Y),Z2))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Z2)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_62_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_63_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_64_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_65_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_66_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_67_Suc__le__D,axiom,
    ! [N: nat,M3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M3))
     => ? [M4: nat] : M3 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_68_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_69_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_70_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_leD
tff(fact_71_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q2) ).

% div_mult2_eq
tff(fact_72_div__le__dividend,axiom,
    ! [M: 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),M),N)),M)) ).

% div_le_dividend
tff(fact_73_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K))) ) ).

% div_le_mono
tff(fact_74_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% of_nat_dvd_iff
tff(fact_75_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_76_Suc__div__le__mono,axiom,
    ! [M: 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),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_77_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_78_Suc__inject,axiom,
    ! [X: nat,Y4: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y4) )
     => ( X = Y4 ) ) ).

% Suc_inject
tff(fact_79_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_80_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_81_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
         => 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),J))) ) ) ).

% of_nat_mono
tff(fact_82_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_83_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_84_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_85_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_86_num_Oexhaust,axiom,
    ! [Y4: num] :
      ( ( Y4 != one2 )
     => ( ! [X22: num] : Y4 != aa(num,num,bit0,X22)
       => ~ ! [X32: num] : Y4 != aa(num,num,bit1,X32) ) ) ).

% num.exhaust
tff(fact_87_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_88_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_89_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_90_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_91_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_92_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_93_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_94_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: 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),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_95_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_96_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_97_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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),times_times(A),A2),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_mult_iff
tff(fact_98_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_99_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_100_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_101_div2__even__ext__nat,axiom,
    ! [X: nat,Y4: 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),Y4),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))),Y4)) )
       => ( X = Y4 ) ) ) ).

% div2_even_ext_nat
tff(fact_102_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(1)
tff(fact_103_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ) ).

% div_dvd_div
tff(fact_104_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),C2)) ) ) ) ).

% dvd_trans
tff(fact_105_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),A2)) ) ).

% dvd_refl
tff(fact_106_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X4: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),S))
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),Z3)) )
       => ? [Y: real] :
            ( ! [X4: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),S))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),Y)) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),Z3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),Z3)) ) ) ) ) ).

% complete_real
tff(fact_107_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ~ ! [K2: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_108_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ).

% dvdI
tff(fact_109_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
        <=> ? [K3: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_110_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult
tff(fact_111_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult2
tff(fact_112_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: 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),A2),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2)) ) ) ).

% dvd_mult_left
tff(fact_113_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ).

% dvd_triv_left
tff(fact_114_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ).

% mult_dvd_mono
tff(fact_115_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: 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),A2),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ).

% dvd_mult_right
tff(fact_116_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))) ) ).

% dvd_triv_right
tff(fact_117_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( 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),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_118_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_119_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: 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),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_120_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: 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)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_121_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: 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),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_122_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: 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),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_123_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_124_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: 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),A2),C2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ).

% dvd_mult_imp_div
tff(fact_125_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( 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),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_126_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_127_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_128_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)))) ) ).

% even_numeral
tff(fact_129_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3) ) ) ).

% evenE
tff(fact_130_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)))) ) ).

% odd_numeral
tff(fact_131_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) != 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_132_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) != 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_133_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_134_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            <=> 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),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_135_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ).

% times_divide_eq_left
tff(fact_136_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_137_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_138_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_139_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_140_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_141_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_142_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ).

% int_dvd_int_iff
tff(fact_143_real__divide__square__eq,axiom,
    ! [R2: real,A2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R2),R2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R2) ).

% real_divide_square_eq
tff(fact_144_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_145_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,V))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_146_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_147_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% subset_decode_imp_le
tff(fact_148_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% zle_int
tff(fact_149_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_150_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_151_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ).

% divide_divide_times_eq
tff(fact_152_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),W)) ) ).

% times_divide_times_eq
tff(fact_153_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_154_verit__eq__simplify_I9_J,axiom,
    ! [X33: num,Y32: num] :
      ( ( aa(num,num,bit1,X33) = aa(num,num,bit1,Y32) )
    <=> ( X33 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_155_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa(num,num,bit0,X2) = aa(num,num,bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_156_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% dual_order.refl
tff(fact_157_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_158_cnt__non__neg,axiom,
    ! [T2: vEBT_VEBT] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2))) ).

% cnt_non_neg
tff(fact_159_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_160_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_161_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_162_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_163_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one2 != aa(num,num,bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_164_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X33: num] : aa(num,num,bit0,X2) != aa(num,num,bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_165_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_166_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_167_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_168_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_169_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_170_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_171_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_172_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_173_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_174_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_175_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_176_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_177_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_178_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A2))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_179_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_180_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_181_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A2)) ).

% bot_nat_0.extremum
tff(fact_182_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_183_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_184_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_185_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_186_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_187_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_188_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_189_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_190_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_191_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_mult_mult1_if
tff(fact_192_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_193_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_194_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_195_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_196_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_197_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_198_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_199_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_200_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_201_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_202_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_203_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_204_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_205_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_206_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_207_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_208_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_209_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_210_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_211_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A2 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_212_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_213_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_214_one__le__mult__iff,axiom,
    ! [M: 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),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
        & 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_215_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),X))
      | ( A2 = X )
      | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A2)) ) ).

% verit_la_generic
tff(fact_216_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_217_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_218_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_219_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_220_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_221_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_222_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_223_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A2))
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_224_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_225_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_226_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_227_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_228_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_229_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_230_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_231_old_Onat_Oexhaust,axiom,
    ! [Y4: nat] :
      ( ( Y4 != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y4 != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_232_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_233_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),zero_zero(nat)))
     => ( ! [Y: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y)))
       => ( ! [X3: nat,Y: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),Y))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).

% diff_induct
tff(fact_234_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_235_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_236_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_237_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_238_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M4: nat] : N = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_239_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_240_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_241_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_242_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_243_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_244_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_245_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_246_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_247_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono
tff(fact_248_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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)),A2))
             => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono'
tff(fact_249_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: 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),A2),A2))) ) ).

% zero_le_square
tff(fact_250_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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),A2),B2))) ) ) ).

% split_mult_pos_le
tff(fact_251_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_252_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_253_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono
tff(fact_254_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono_neg
tff(fact_255_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono
tff(fact_256_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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_257_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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),A2),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_258_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_259_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_260_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_261_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_262_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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_263_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_264_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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_265_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_right_mono
tff(fact_266_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & 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),A2),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_267_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => 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),Y4))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_268_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_269_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => 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),Y4)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_270_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_271_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_272_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y4: A,Z: A,X: A,W: A] :
          ( ( Y4 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4) = 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),Y4) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_273_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_274_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_275_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_276_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_277_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_278_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_279_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_280_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_281_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_282_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% verit_comp_simplify1(2)
tff(fact_283_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_284_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A,Z: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
             => ~ 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),Z))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y4)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y4))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z))
                   => ~ 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),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_285_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( ( X = Y4 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_286_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = 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_eq(A),A2),C2)) ) ) ) ).

% ord_eq_le_trans
tff(fact_287_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% ord_le_eq_trans
tff(fact_288_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
           => ( X = Y4 ) ) ) ) ).

% order_antisym
tff(fact_289_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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_eq(A),A2),C2)) ) ) ) ).

% order.trans
tff(fact_290_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% order_trans
tff(fact_291_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
         => ( ! [A4: A,B3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P,B3),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ).

% linorder_wlog
tff(fact_292_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_293_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_294_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( 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),A2)) ) ) ) ).

% dual_order.trans
tff(fact_295_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_296_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funD
tff(fact_297_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funE
tff(fact_298_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G)) ) ) ).

% le_funI
tff(fact_299_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
        <=> ! [X5: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X5)),aa(A,B,G,X5))) ) ) ).

% le_fun_def
tff(fact_300_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_301_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_subst1
tff(fact_302_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,X3)),aa(A,C,F2,Y))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_subst2
tff(fact_303_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A] :
          ( ( X = Y4 )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% order_eq_refl
tff(fact_304_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% verit_la_disequality
tff(fact_305_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ).

% linorder_linear
tff(fact_306_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_307_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),C2)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_308_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ).

% linorder_le_cases
tff(fact_309_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
          <=> ( X = Y4 ) ) ) ) ).

% order_antisym_conv
tff(fact_310_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_311_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_312_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_313_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_314_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_315_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: 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),A2),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),A2),C2)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_316_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_317_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_318_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_319_int__if,axiom,
    ! [P: bool,A2: nat,B2: nat] :
      ( ( pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A2,B2)) = aa(nat,int,semiring_1_of_nat(int),A2) ) )
      & ( ~ pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A2,B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).

% int_if
tff(fact_320_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_321_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_322_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_323_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    vEBT_V8646137997579335489_i_l_d(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_324_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_325_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one2 != aa(num,num,bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_326_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_327_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_328_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: 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(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_329_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: 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(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)))
        <=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_330_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: 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(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_331_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
       => ( A3 = B4 ) ) ) ).

% subset_antisym
tff(fact_332_subsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% subsetI
tff(fact_333_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_334_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_335_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)),aa(int,int,aa(nat,fun(int,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_336_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)),aa(int,int,aa(nat,fun(int,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_337_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)),aa(int,int,aa(nat,fun(int,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_338_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_339_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_340_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_341_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_342_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_343_Suc__diff__diff,axiom,
    ! [M: 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,M)),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),M),N)),K) ).

% Suc_diff_diff
tff(fact_344_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_Suc_Suc
tff(fact_345_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_346_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_347_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_348_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: 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),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_349_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_350_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_351_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_352_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% diff_is_0_eq
tff(fact_353_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_354_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_355_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_356_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_357_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_358_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_359_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_right_mono
tff(fact_360_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => 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),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_left_mono
tff(fact_361_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))) ) ) ) ).

% diff_mono
tff(fact_362_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_363_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_364_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib'
tff(fact_365_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ).

% left_diff_distrib'
tff(fact_366_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib
tff(fact_367_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_368_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% diff_divide_distrib
tff(fact_369_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% dvd_diff_commute
tff(fact_370_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y4: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4))
         => ( 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),Y4),Z))) ) ) ) ).

% dvd_diff
tff(fact_371_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_372_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_373_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_374_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( 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),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_375_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( 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),M),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),M),N)) ) ) ) ).

% le_diff_iff
tff(fact_376_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( 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),M),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),M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_377_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_378_diff__le__self,axiom,
    ! [M: 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),M),N)),M)) ).

% diff_le_self
tff(fact_379_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),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),A2)),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),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_380_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M))) ) ).

% diff_le_mono2
tff(fact_381_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_382_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_383_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
     => ( 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),M),N))) ) ) ).

% dvd_diff_nat
tff(fact_384_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_385_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_386_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_387_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_388_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),N),zero_zero(extended_enat)))
    <=> ( N = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_389_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),zero_zero(extended_enat)),N)) ).

% i0_lb
tff(fact_390_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K)) ).

% unset_bit_less_eq
tff(fact_391_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))) ).

% set_bit_greater_eq
tff(fact_392_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ) ).

% Suc_diff_le
tff(fact_393_dvd__diffD,axiom,
    ! [K: nat,M: 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),M),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),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ) ).

% dvd_diffD
tff(fact_394_dvd__diffD1,axiom,
    ! [K: nat,M: 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),M),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_395_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_396_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_397_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_398_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_399_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( 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),M),N))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_400_zdvd__mult__cancel,axiom,
    ! [K: int,M: 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),M)),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),M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_401_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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)),Y4) = 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),Y4),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_402_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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),Y4),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)),Y4)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_403_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y4: A,Z: A,X: A,W: A] :
          ( ( Y4 != 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),Y4)),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),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_404_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_405_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_406_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.left_commute
tff(fact_407_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% mult.commute
tff(fact_408_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_409_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_410_in__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4)) ) ) ).

% in_mono
tff(fact_411_subsetD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% subsetD
tff(fact_412_equalityE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ) ).

% equalityE
tff(fact_413_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
    <=> ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4)) ) ) ).

% subset_eq
tff(fact_414_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% equalityD1
tff(fact_415_equalityD2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ).

% equalityD2
tff(fact_416_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
    <=> ! [T3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T3),B4)) ) ) ).

% subset_iff
tff(fact_417_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A3)) ).

% subset_refl
tff(fact_418_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))) ) ).

% Collect_mono
tff(fact_419_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3)) ) ) ).

% subset_trans
tff(fact_420_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ) ).

% set_eq_subset
tff(fact_421_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)))
    <=> ! [X5: A] :
          ( pp(aa(A,bool,P,X5))
         => pp(aa(A,bool,Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_422_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: 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),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_423_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,Z: A,X: A,W: A] :
          ( ( Y4 != 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),Y4)),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),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_424_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    vEBT_V8346862874174094_d_u_p(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_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_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X3: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A2))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)) = D3 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_427_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_428_minus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B4: 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)),A3),B4),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X)),aa(A,B,B4,X)) ) ).

% minus_apply
tff(fact_429_buildup__build__time,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_V8346862874174094_d_u_p(N)),vEBT_V8646137997579335489_i_l_d(N))) ).

% buildup_build_time
tff(fact_430_zdvd__mono,axiom,
    ! [K: int,M: int,T2: int] :
      ( ( K != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),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),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),T2))) ) ) ).

% zdvd_mono
tff(fact_431_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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)),A2)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_432_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_433_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_434_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_435_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),zero_zero(nat)))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_436_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_437_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_438_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_439_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_440_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% numeral_less_iff
tff(fact_441_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_442_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_443_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_444_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_445_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_446_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_447_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_eq
tff(fact_448_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_449_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_450_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_451_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_452_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_453_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_454_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: 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),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_455_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_456_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_457_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_458_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_459_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_460_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_461_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_462_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_463_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_464_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_465_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_466_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_467_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_468_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_469_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_470_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_471_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_472_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)),one_one(A))) ) ) ) ).

% unit_div
tff(fact_473_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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)),A2)),one_one(A))) ) ) ).

% unit_div_1_unit
tff(fact_474_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_475_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_iff
tff(fact_476_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_477_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_478_zero__less__diff,axiom,
    ! [N: nat,M: 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),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% zero_less_diff
tff(fact_479_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_480_nat__0__less__mult__iff,axiom,
    ! [M: 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),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_481_mult__less__cancel2,axiom,
    ! [M: 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),M),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),M),N)) ) ) ).

% mult_less_cancel2
tff(fact_482_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_483_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_484_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_485_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_486_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_487_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_488_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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)),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_489_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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)),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_divide_1_iff
tff(fact_490_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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)),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_divide_1_iff
tff(fact_491_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_492_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_493_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_494_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_495_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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)),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_496_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_497_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: 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))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_498_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_499_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_500_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_501_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_502_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_503_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A2: 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),A2),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),A2),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_504_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_505_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_506_mult__le__cancel2,axiom,
    ! [M: 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),M),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),M),N)) ) ) ).

% mult_le_cancel2
tff(fact_507_div__mult__self1__is__m,axiom,
    ! [N: nat,M: 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),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_508_div__mult__self__is__m,axiom,
    ! [N: nat,M: 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),M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_509_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_510_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_511_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_512_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_513_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_514_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_515_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_516_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_517_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_518_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_519_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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),A2),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_520_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_521_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_522_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_523_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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)))
              & ( A2 != 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)),A2)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_524_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),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)))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_525_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_526_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( N = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_527_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_528_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X4)) ) ).

% linordered_field_no_lb
tff(fact_529_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X_1)) ) ).

% linordered_field_no_ub
tff(fact_530_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_531_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% nat_neq_iff
tff(fact_532_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_533_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_534_less__not__refl3,axiom,
    ! [S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T2))
     => ( S2 != T2 ) ) ).

% less_not_refl3
tff(fact_535_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X3)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct
tff(fact_536_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_537_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_538_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
              & ~ pp(aa(nat,bool,P,M2)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_539_linorder__neqE__nat,axiom,
    ! [X: nat,Y4: nat] :
      ( ( X != Y4 )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y4))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y4),X)) ) ) ).

% linorder_neqE_nat
tff(fact_540_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X3)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct_rule
tff(fact_541_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V2: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P,X3))
         => ? [Y3: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3)))
              & ~ pp(aa(A,bool,P,Y3)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_542_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: A] :
          ( ( X != Y4 )
         => ( ~ 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),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_543_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% verit_comp_simplify1(1)
tff(fact_544_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ).

% lt_ex
tff(fact_545_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_546_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ? [Z2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y4)) ) ) ) ).

% dense
tff(fact_547_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ( X != Y4 ) ) ) ).

% less_imp_neq
tff(fact_548_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order.asym
tff(fact_549_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = 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),A2),C2)) ) ) ) ).

% ord_eq_less_trans
tff(fact_550_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% ord_less_eq_trans
tff(fact_551_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X3))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% less_induct
tff(fact_552_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y4: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
          <=> ( X = Y4 ) ) ) ) ).

% antisym_conv3
tff(fact_553_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ( ( X != Y4 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ) ) ).

% linorder_cases
tff(fact_554_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% dual_order.asym
tff(fact_555_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% dual_order.irrefl
tff(fact_556_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
        <=> ? [N4: A] :
              ( pp(aa(A,bool,P,N4))
              & ! [M5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M5),N4))
                 => ~ pp(aa(A,bool,P,M5)) ) ) ) ) ).

% exists_least_iff
tff(fact_557_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A4),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
         => ( ! [A4: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A4),A4))
           => ( ! [A4: A,B3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P,B3),A4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_558_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)) ) ) ) ).

% order.strict_trans
tff(fact_559_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,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),X))
            | ( X = Y4 ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_560_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2)) ) ) ) ).

% dual_order.strict_trans
tff(fact_561_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_562_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_563_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( ( X != Y4 )
         => ( ~ 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),X)) ) ) ) ).

% linorder_neqE
tff(fact_564_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,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),X)) ) ) ).

% order_less_asym
tff(fact_565_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( ( X != Y4 )
        <=> ( 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),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_566_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order_less_asym'
tff(fact_567_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A,Z: 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),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_trans
tff(fact_568_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_569_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,A2)),C2)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_570_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_571_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_subst1
tff(fact_572_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X3)),aa(A,C,F2,Y))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_less_subst2
tff(fact_573_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,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),X)) ) ) ).

% order_less_not_sym
tff(fact_574_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A,P: bool] :
          ( 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),X))
           => pp(P) ) ) ) ).

% order_less_imp_triv
tff(fact_575_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
          | ( X = Y4 )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ) ).

% linorder_less_linear
tff(fact_576_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ( X != Y4 ) ) ) ).

% order_less_imp_not_eq
tff(fact_577_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ( Y4 != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_578_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,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),X)) ) ) ).

% order_less_imp_not_less
tff(fact_579_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_580_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_581_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_582_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_583_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
         => ( 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),M),N))) ) ) ) ).

% less_1_mult
tff(fact_584_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,M: 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,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_585_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_586_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_587_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_588_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X4)) ) ) ).

% minf(8)
tff(fact_589_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),T2)) ) ) ).

% minf(6)
tff(fact_590_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X4)) ) ) ).

% pinf(8)
tff(fact_591_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X4))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),T2)) ) ) ).

% pinf(6)
tff(fact_592_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: 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),A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_593_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: 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),A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_594_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_595_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_596_div__less__dividend,axiom,
    ! [N: nat,M: 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)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_597_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B5: B,A5: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B5),A5))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A5),B5)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_598_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ).

% leD
tff(fact_599_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,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_eq(A),Y4),X)) ) ) ).

% leI
tff(fact_600_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_601_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,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_eq(A),X),Y4))
          <=> ( X = Y4 ) ) ) ) ).

% antisym_conv1
tff(fact_602_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
          <=> ( X = Y4 ) ) ) ) ).

% antisym_conv2
tff(fact_603_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y4: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z)) ) ) ).

% dense_ge
tff(fact_604_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y4: A,Z: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z)) ) ) ).

% dense_le
tff(fact_605_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,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_eq(A),X),Y4))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ) ).

% less_le_not_le
tff(fact_606_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y4: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ).

% not_le_imp_less
tff(fact_607_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_608_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_609_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_610_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_611_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% order.strict_iff_not
tff(fact_612_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y4: 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),Y4),W2)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Z)) ) ) ) ).

% dense_ge_bounded
tff(fact_613_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y4: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
         => ( ! [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),Y4))
                 => 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),Y4),Z)) ) ) ) ).

% dense_le_bounded
tff(fact_614_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_615_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_616_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( 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),A2)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_617_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_618_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_619_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% order.strict_implies_order
tff(fact_620_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% dual_order.strict_implies_order
tff(fact_621_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
            | ( X = Y4 ) ) ) ) ).

% order_le_less
tff(fact_622_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,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_eq(A),X),Y4))
            & ( X != Y4 ) ) ) ) ).

% order_less_le
tff(fact_623_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ) ).

% linorder_not_le
tff(fact_624_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,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_eq(A),Y4),X)) ) ) ).

% linorder_not_less
tff(fact_625_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,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_eq(A),X),Y4)) ) ) ).

% order_less_imp_le
tff(fact_626_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( A2 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_627_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_628_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_le_less_trans
tff(fact_629_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A,Z: 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_eq(A),Y4),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_le_trans
tff(fact_630_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_631_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,X3)),aa(A,C,F2,Y))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_632_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_633_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X3)),aa(A,C,F2,Y))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_634_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ) ).

% linorder_le_less_linear
tff(fact_635_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
            | ( X = Y4 ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_636_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_637_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_638_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_639_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_640_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_641_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_642_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y4: 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),Y4))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),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),Y4)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y4))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_643_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_strict_right_mono
tff(fact_644_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_645_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less
tff(fact_646_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))) ) ) ) ).

% diff_strict_mono
tff(fact_647_double__diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_648_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B4: 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)),A3),B4)),A3)) ).

% Diff_subset
tff(fact_649_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),D4: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),B4))
       => 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)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),D4))) ) ) ).

% Diff_mono
tff(fact_650_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_651_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) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_652_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_lessD
tff(fact_653_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))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_654_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_655_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_656_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_657_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_658_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_659_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_660_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_661_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
    <=> ? [M6: nat] :
          ( ( M = aa(nat,nat,suc,M6) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_662_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_663_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_SucD
tff(fact_664_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_665_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),aa(nat,nat,suc,I3)))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_666_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I3: nat] :
            ( ( J = 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),J))
             => ( 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_667_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_668_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_669_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_670_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_671_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_672_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_673_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_674_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_675_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))
             => ? [M2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
                  & ~ pp(aa(nat,bool,P,M2)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_676_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V2,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X3)))
           => ( ~ pp(aa(A,bool,P,X3))
             => ? [Y3: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3)))
                  & ~ pp(aa(A,bool,P,Y3)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_677_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_678_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_679_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_680_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_681_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,I3)),aa(nat,nat,F2,J2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_682_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% le_neq_implies_less
tff(fact_683_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_or_eq_imp_le
tff(fact_684_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_685_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_imp_le_nat
tff(fact_686_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_687_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N2: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M4)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_688_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),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),M))) ) ) ).

% diff_less_mono2
tff(fact_689_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_690_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_691_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2)) ) ) ).

% unit_imp_dvd
tff(fact_692_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),one_one(A)),A2)) ) ).

% one_dvd
tff(fact_693_zdvd__zdiffD,axiom,
    ! [K: int,M: 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),M),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),M)) ) ) ).

% zdvd_zdiffD
tff(fact_694_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_695_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_696_dvd__pos__nat,axiom,
    ! [N: nat,M: 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),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_697_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_698_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: 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),A2)),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),A2),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)),A2)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_699_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_700_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),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)),A2)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_701_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_702_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: 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),A2)),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),A2),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)),A2)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_703_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_704_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),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)),A2)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_705_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: A] :
          ( ! [Z2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),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),Z2),X)),Y4)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% field_le_mult_one_interval
tff(fact_706_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: 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),A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_707_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: 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),A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_708_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_709_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: 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,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_710_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_711_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_712_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_713_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_714_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_715_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_716_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_717_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_718_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & 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),A2),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_719_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_720_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_721_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_722_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_723_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & 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),A2),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_724_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: 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),A2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_725_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: 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),A2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_726_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_727_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_728_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_729_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_730_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),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),A2)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_731_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_732_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono
tff(fact_733_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),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),A2),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),A2)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_734_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_735_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_736_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4))) ) ) ) ).

% divide_neg_neg
tff(fact_737_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_738_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_739_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => 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),Y4))) ) ) ) ).

% divide_pos_pos
tff(fact_740_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & 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),A2),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_741_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),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),A2),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),A2)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_742_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & 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),A2),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_743_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono
tff(fact_744_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_745_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_746_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_747_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_748_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_749_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_750_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_751_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M4: nat] : N = aa(nat,nat,suc,M4) ) ).

% gr0_implies_Suc
tff(fact_752_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_753_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_754_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_755_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_756_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_757_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_Suc_eq_le
tff(fact_758_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_759_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_lessD
tff(fact_760_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [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),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% inc_induct
tff(fact_761_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( 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),J))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_762_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_eq
tff(fact_763_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_764_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_765_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_766_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_767_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_768_diff__less__Suc,axiom,
    ! [M: 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),M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_769_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ).

% Suc_diff_Suc
tff(fact_770_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: 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)),M)),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),M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_771_diff__less,axiom,
    ! [N: nat,M: 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)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ) ) ).

% diff_less
tff(fact_772_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_773_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: 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),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_774_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( 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),J))) ) ) ).

% mult_less_mono2
tff(fact_775_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( 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),J),K))) ) ) ).

% mult_less_mono1
tff(fact_776_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: 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),A2),B2)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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_777_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),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),A2),C2)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_778_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_779_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),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),A2),C2)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_780_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_781_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_782_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_783_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),A2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2))) ) ) ).

% diff_less_mono
tff(fact_784_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( 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),M),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),M),N)) ) ) ) ).

% less_diff_iff
tff(fact_785_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_786_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_787_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_788_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),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),A2),C2)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_789_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_790_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_791_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_792_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_minus_self
tff(fact_793_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),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),M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_794_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),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),M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_795_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_796_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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))
              & ( A2 != 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)),A2)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_797_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),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),A2)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_798_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),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),A2),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),A2)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_799_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_800_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_801_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),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),A2)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_802_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_803_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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)),A2))
             => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_804_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: 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),A2),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),A2),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),A2)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_805_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_806_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_807_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_808_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_809_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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)),A2))
             => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_810_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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)),A2))
             => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_811_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,X: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => ( 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),Y4),W))) ) ) ) ) ) ).

% frac_le
tff(fact_812_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4))
           => ( 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),Y4),W))) ) ) ) ) ) ).

% frac_less
tff(fact_813_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4))
           => ( 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),Y4),W))) ) ) ) ) ) ).

% frac_less2
tff(fact_814_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),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),A2),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),A2)) ) ) ) ) ).

% divide_le_cancel
tff(fact_815_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_816_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => 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),Y4))) ) ) ) ).

% divide_nonneg_pos
tff(fact_817_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),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),Y4))) ) ) ) ).

% divide_nonpos_neg
tff(fact_818_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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)),Y4))
           => 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),Y4)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_819_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_820_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: 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),A2))
           => 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),A2),B2))) ) ) ) ).

% div_positive
tff(fact_821_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
        <=> ( ( 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),A2),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),A2),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)),A2)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_822_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),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),A2),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),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_823_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_824_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_825_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_826_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_827_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y4))
         => ( 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),Y4)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_828_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y4)),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),Y4))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_829_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( 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),A2),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),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_830_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),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),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_831_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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)),Y4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),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),Y4),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_832_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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)),Y4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),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),Y4)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_833_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_834_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: 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)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2)) ) ) ) ).

% mult_left_le
tff(fact_835_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_836_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_837_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_838_n__less__n__mult__m,axiom,
    ! [N: nat,M: 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))),M))
       => 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),M))) ) ) ).

% n_less_n_mult_m
tff(fact_839_n__less__m__mult__n,axiom,
    ! [N: nat,M: 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))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_840_one__less__mult,axiom,
    ! [N: nat,M: 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))),M))
       => 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),M),N))) ) ) ).

% one_less_mult
tff(fact_841_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_842_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_843_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_844_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_845_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: 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),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_846_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_847_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: 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),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_848_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_849_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B4: fun(A,B),X4: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A3),B4),X4) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X4)),aa(A,B,B4,X4)) ) ).

% fun_diff_def
tff(fact_850_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_851_div__greater__zero__iff,axiom,
    ! [M: 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),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_852_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M))) ) ) ).

% div_le_mono2
tff(fact_853_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_854_dvd__mult__cancel,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_855_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q2)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_856_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: 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),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_857_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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),A2),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))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_858_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: 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))),M)),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),M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_859_conj__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( 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)),X6))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_860_imp__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( 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)),X6))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_861_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_862_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_863_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2)) ) ).

% gcd_nat.strict_implies_order
tff(fact_864_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_865_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_866_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_867_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_868_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_869_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_870_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2))
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_871_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),A2))
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_872_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2)) ) ) ).

% gcd_nat.eq_iff
tff(fact_873_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2)) ) ) ).

% gcd_nat.trans
tff(fact_874_gcd__nat_Orefl,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),A2)) ).

% gcd_nat.refl
tff(fact_875_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2))
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_876_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: 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))),M)),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),M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_877_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
        <=> ( ( 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),A2),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),A2),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)),A2)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_878_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),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),A2),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),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_879_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( 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),A2),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),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_880_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_881_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_882_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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)),A2))
          <=> 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),A2),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_883_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_884_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y4))
         => ( 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),Y4)))
           => 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),Y4)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_885_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y4))
         => ( 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),Y4)),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),Y4))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_886_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),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),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_887_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_888_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_889_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,Z: A,X: A,W: A] :
          ( ( Y4 != 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),Y4)),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),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_890_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_891_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_892_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),one_one(A)))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_893_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_894_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_895_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_896_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% div_if
tff(fact_897_div__geq,axiom,
    ! [N: nat,M: 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),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% div_geq
tff(fact_898_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: 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),Q2)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2))))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_899_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_900_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),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),M))
            | ( 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),M),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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_901_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_902_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: 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),A2),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),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_903_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_904_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_905_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => 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),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% half_gt_zero
tff(fact_906_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),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)),A2)) ) ) ).

% half_gt_zero_iff
tff(fact_907_four__x__squared,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% four_x_squared
tff(fact_908_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_909_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_910_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_911_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_912_le__div__geq,axiom,
    ! [N: nat,M: 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),M))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_913_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q3: 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),Q3)),M))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q3))))
            & pp(aa(nat,bool,P,Q3)) ) ) ) ).

% split_div'
tff(fact_914_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( ( 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),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_915_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))
       => ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M4))
             => 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),M4)),X)),C2)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_916_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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)),A2)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_917_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: 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),A2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% zero_le_odd_power
tff(fact_918_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: 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),A2),N))) ) ) ).

% zero_le_even_power
tff(fact_919_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_920_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_921_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_922_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K2: A] :
              ( pp(aa(A,bool,P,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),K2),D4)))) )
         => ( ! [X3: A,K2: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> 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),K2),D4)))) )
           => ! [X4: A,K4: A] :
                ( ( pp(aa(A,bool,P,X4))
                  | pp(aa(A,bool,Q,X4)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))))
                  | pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_923_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K2: A] :
              ( pp(aa(A,bool,P,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),K2),D4)))) )
         => ( ! [X3: A,K2: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> 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),K2),D4)))) )
           => ! [X4: A,K4: A] :
                ( ( pp(aa(A,bool,P,X4))
                  & pp(aa(A,bool,Q,X4)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))))
                  & pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_924_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
         => ~ ! [X3: A,Y: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y) )
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),A2))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y),B2)) ) ) ) ) ).

% dvd_productE
tff(fact_925_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ? [B6: A,C5: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),C5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B6),B2))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C5),C2)) ) ) ) ).

% division_decomp
tff(fact_926_space__space_H,axiom,
    ! [T2: vEBT_VEBT] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))) ).

% space_space'
tff(fact_927_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_928_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_929_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_930_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_931_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_932_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_933_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_934_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y4: 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)),Y4))
           => ( ( 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y4 ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_935_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: 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),M)),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),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_936_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),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),A2),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_937_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( ( A3 != B4 )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4)) ) ) ).

% psubsetI
tff(fact_938_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4))
       => 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)),A3),B4))) ) ) ).

% DiffI
tff(fact_939_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),A3),B4)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% Diff_iff
tff(fact_940_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ).

% Diff_idemp
tff(fact_941_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N))
    <=> ( N != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_942_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),N)))
     => 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(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_two
tff(fact_943_not__real__square__gt__zero,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),times_times(real),X),X)))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_944_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_945_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(78)
tff(fact_946_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one2)) ).

% semiring_norm(75)
tff(fact_947_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(80)
tff(fact_948_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_949_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_950_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,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_951_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_952_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_953_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_954_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_955_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_956_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_957_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),N) ) ).

% of_nat_power
tff(fact_958_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_959_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_960_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_961_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_962_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% power_mult_numeral
tff(fact_963_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit0,N))) ).

% semiring_norm(76)
tff(fact_964_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(81)
tff(fact_965_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit1,N))) ).

% semiring_norm(77)
tff(fact_966_enat__ord__number_I2_J,axiom,
    ! [M: 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),M)),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),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(2)
tff(fact_967_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_968_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_969_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_970_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(74)
tff(fact_971_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(79)
tff(fact_972_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: 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),M)),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),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_973_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y4: 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),Y4)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y4)) ) ) ) ).

% power_increasing_iff
tff(fact_974_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_975_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_976_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_977_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_978_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_979_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y4: nat,X: num,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y4) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y4 = 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_980_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,N: nat,Y4: 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),Y4) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = Y4 ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_981_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_982_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),A3),B4)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% DiffE
tff(fact_983_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),A3),B4)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% DiffD1
tff(fact_984_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),A3),B4)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ).

% DiffD2
tff(fact_985_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => ? [B3: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))) ) ).

% psubset_imp_ex_mem
tff(fact_986_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_987_real__arch__pow,axiom,
    ! [X: real,Y4: 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),Y4),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2))) ) ).

% real_arch_pow
tff(fact_988_real__arch__pow__inv,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4))
     => ( 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)),Y4)) ) ) ).

% real_arch_pow_inv
tff(fact_989_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_990_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_991_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_992_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ) ).

% psubsetE
tff(fact_993_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_994_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% psubset_imp_subset
tff(fact_995_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% psubset_subset_trans
tff(fact_996_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
        & ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ) ).

% subset_not_subset_eq
tff(fact_997_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B4),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% subset_psubset_trans
tff(fact_998_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_999_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_1000_less__eq__real__def,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4))
        | ( X = Y4 ) ) ) ).

% less_eq_real_def
tff(fact_1001_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F2),G))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G),F2)) ) ) ) ).

% less_fun_def
tff(fact_1002_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_1003_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N)))
    <=> ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),M))
        & pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_1004_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),N),zero_zero(extended_enat))) ).

% not_iless0
tff(fact_1005_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1006_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ? [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)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1007_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(2)
tff(fact_1008_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
     => ( 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),J))) ) ) ).

% zmult_zless_mono2
tff(fact_1009_plusinfinity,axiom,
    ! [D2: int,P2: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P2,X3))
          <=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z3),X3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P2,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P2,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1010_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))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P1,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1011_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: 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),A2),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1012_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: 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),A2),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1013_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),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),A2),B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1014_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).

% zdvd_not_zless
tff(fact_1015_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1016_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: 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)),A2))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X3),N) = A2 )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3))
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y3),N) = A2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1017_realpow__pos__nth,axiom,
    ! [N: nat,A2: 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)),A2))
       => ? [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) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1018_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).

% int_ops(6)
tff(fact_1019_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))
     => ( ! [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),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X4: int] :
              ( pp(aa(int,bool,P,X4))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1020_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y3: real] :
        ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),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_1021_zdiv__mono1,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A5))
     => ( 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),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2))) ) ) ).

% zdiv_mono1
tff(fact_1022_zdiv__mono2,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B5),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),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B5))) ) ) ) ).

% zdiv_mono2
tff(fact_1023_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_1024_zdiv__mono1__neg,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A5))
     => ( 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),A5),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1025_zdiv__mono2__neg,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B5),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),A2),B5)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1026_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_1027_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_1028_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( 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),A2),B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1029_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),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),A2),B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1030_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_1031_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: 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),A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1032_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: 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),A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1033_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( 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),A2),B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1034_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_1035_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_1036_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_1037_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
     => ( 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)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1038_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_1039_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_1040_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( 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),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1041_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_1042_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y4: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_1043_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_mult_distrib
tff(fact_1044_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_commutes
tff(fact_1045_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_divide
tff(fact_1046_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y4: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y4),N))) ) ) ).

% dvd_power_same
tff(fact_1047_nat__power__less__imp__less,axiom,
    ! [I: nat,M: 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),M)),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),M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_1048_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),N) ) ).

% power_mult
tff(fact_1049_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_1050_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => 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),A2),N))) ) ) ).

% zero_le_power
tff(fact_1051_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => 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),A2),N))) ) ) ).

% zero_less_power
tff(fact_1052_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => 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),A2),N))) ) ) ).

% one_le_power
tff(fact_1053_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y4: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4) = 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),Y4),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1054_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),A2) ) ).

% power_Suc2
tff(fact_1055_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_Suc
tff(fact_1056_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_one_over
tff(fact_1057_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1058_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ) ) ).

% div_power
tff(fact_1059_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_1060_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_1061_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y4: A,N: nat,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y4),M))) ) ) ) ).

% dvd_power_le
tff(fact_1062_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat,B2: A,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_1063_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% le_imp_power_dvd
tff(fact_1064_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_1065_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_1066_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: 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),A2),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,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),A2),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_1067_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1068_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1069_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1070_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_1071_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => 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),A2),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1072_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N5))) ) ) ) ).

% power_increasing
tff(fact_1073_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_1074_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_1075_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: 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),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_1076_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: 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),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_1077_self__le__ge2__pow,axiom,
    ! [K: nat,M: 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),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M))) ) ).

% self_le_ge2_pow
tff(fact_1078_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: 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),A2),N)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1079_power__dvd__imp__le,axiom,
    ! [I: nat,M: 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),M)),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),M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_1080_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% power_Suc_less
tff(fact_1081_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),aa(nat,nat,suc,N))),A2)) ) ) ) ).

% power_Suc_le_self
tff(fact_1082_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1083_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1084_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_decreasing
tff(fact_1085_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1086_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_1087_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A2: 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)),A2))
           => ( 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),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1088_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( 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))
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1089_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_1090_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_1091_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_1092_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => ( 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% self_le_power
tff(fact_1093_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)),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),Y4),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_commute
tff(fact_1094_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( 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),A2),N))) ) ) ) ).

% one_less_power
tff(fact_1095_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: 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),M)),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),M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_1096_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_1097_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_1098_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ) ).

% power_diff
tff(fact_1099_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power_even_eq
tff(fact_1100_diff__le__diff__pow,axiom,
    ! [K: nat,M: 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),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))) ) ).

% diff_le_diff_pow
tff(fact_1101_dvd__power__iff__le,axiom,
    ! [K: nat,M: 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),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_1102_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y4: 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),Y4),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)),Y4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ).

% power2_le_imp_le
tff(fact_1103_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y4: 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),Y4),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)),Y4))
             => ( X = Y4 ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1104_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% zero_le_power2
tff(fact_1105_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_1106_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( 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),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_1107_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P3: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1108_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A2: 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),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% power_minus_mult
tff(fact_1109_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y4: 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),Y4),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)),Y4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ) ).

% power2_less_imp_less
tff(fact_1110_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),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_1111_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_1112_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1113_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_1114_space__2__pow__bound,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),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,aa(num,num,bit1,one2))))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N)),one_one(real))))) ) ).

% space_2_pow_bound
tff(fact_1115_cnt__bound,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))))))) ) ).

% cnt_bound
tff(fact_1116_space__bound,axiom,
    ! [T2: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( U = 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_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),U))) ) ) ).

% space_bound
tff(fact_1117_cnt__bound_H,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N)),one_one(real))))) ) ).

% cnt_bound'
tff(fact_1118_space_H__bound,axiom,
    ! [T2: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( U = 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_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),U))) ) ) ).

% space'_bound
tff(fact_1119_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_1120_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_1121_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2) ).

% pow_sum
tff(fact_1122_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_1123_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_1124_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_1125_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_1126_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_1127_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_1128_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: 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),M))))
     => 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))),M))) ) ).

% high_bound_aux
tff(fact_1129_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ).

% add_le_cancel_left
tff(fact_1130_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ).

% add_le_cancel_right
tff(fact_1131_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_1132_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_1133_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y4: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4) )
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1134_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1135_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_1136_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_1137_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_1138_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_1139_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_1140_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_1141_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ).

% add_less_cancel_left
tff(fact_1142_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ).

% add_less_cancel_right
tff(fact_1143_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_1144_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_1145_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_1146_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1147_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_1148_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1149_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_1150_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_1151_high__inv,axiom,
    ! [X: nat,N: nat,Y4: 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),Y4),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) = Y4 ) ) ).

% high_inv
tff(fact_1152_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1153_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1154_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_1155_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_1156_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_1157_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_1158_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: 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),M)),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),M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_1159_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1160_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: 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),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1161_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: 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),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1162_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_1163_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_1164_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: 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),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_1165_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: 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),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_1166_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: 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),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_1167_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: 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),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_1168_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_1169_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_1170_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1171_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: 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),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1172_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y4: 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),Y4),Y4)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1173_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1174_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1175_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1176_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_1177_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_1178_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1179_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1180_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_1181_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_1182_add__gr__0,axiom,
    ! [M: 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),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_1183_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_1184_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1185_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1186_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1187_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_1188_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1189_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1190_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1191_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1192_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_1193_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_1194_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_1195_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1196_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1197_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_1198_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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),plus_plus(A),A2),B2)))
        <=> ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% odd_add
tff(fact_1199_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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),plus_plus(A),A2),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_add
tff(fact_1200_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_1201_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_1202_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = M ).

% add_self_div_2
tff(fact_1203_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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),A2),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))),A2)) ) ) ).

% even_plus_one_iff
tff(fact_1204_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1205_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: 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),A2),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),A2),B2))) ) ) ).

% even_diff
tff(fact_1206_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_1207_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1208_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1209_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1210_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1211_even__diff__nat,axiom,
    ! [M: 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),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),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),M),N))) ) ) ).

% even_diff_nat
tff(fact_1212_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1213_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1214_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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))),A2))
         => ( 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)),A2)),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),A2),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_1215_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% psubsetD
tff(fact_1216_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B4),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% psubset_trans
tff(fact_1217_enat__less__induct,axiom,
    ! [P: fun(extended_enat,bool),N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),M2),N2))
             => pp(aa(extended_enat,bool,P,M2)) )
         => pp(aa(extended_enat,bool,P,N2)) )
     => pp(aa(extended_enat,bool,P,N)) ) ).

% enat_less_induct
tff(fact_1218_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1219_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1220_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_1221_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_1222_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_1223_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_1224_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_1225_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% add.commute
tff(fact_1226_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.left_commute
tff(fact_1227_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_1228_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_1229_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_1230_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & ( 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),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1231_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,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),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1232_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & 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),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1233_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_mono
tff(fact_1234_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_left_mono
tff(fact_1235_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% less_eqE
tff(fact_1236_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_right_mono
tff(fact_1237_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ? [C6: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C6) ) ) ).

% le_iff_add
tff(fact_1238_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_1239_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_1240_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_1241_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_1242_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_1243_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_1244_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & 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),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1245_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,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),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1246_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & ( 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),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1247_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_strict_mono
tff(fact_1248_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_strict_left_mono
tff(fact_1249_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_strict_right_mono
tff(fact_1250_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: 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),A2)),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),A2),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_1251_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: 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),A2),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),A2),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_1252_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_1253_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_1254_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_1255_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% distrib_left
tff(fact_1256_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_1257_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: 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),A2),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),A2),B2)),E2)),C2) ) ).

% combine_common_factor
tff(fact_1258_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1259_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1260_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1261_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1262_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1263_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1264_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1265_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1266_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1267_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2)) ) ).

% add_diff_add
tff(fact_1268_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_1269_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),C2)) ) ) ) ).

% dvd_add_right_iff
tff(fact_1270_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_1271_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).

% dvd_add
tff(fact_1272_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_1273_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_1274_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_1275_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B3),A4)) )
     => ( ! [A4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),zero_zero(nat)))
       => ( ! [A4: nat,B3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A2),B2)) ) ) ) ).

% Euclid_induct
tff(fact_1276_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_1277_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_1278_add__lessD1,axiom,
    ! [I: nat,J: 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),J)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).

% add_lessD1
tff(fact_1279_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( 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),J),L))) ) ) ).

% add_less_mono
tff(fact_1280_not__add__less1,axiom,
    ! [I: nat,J: 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),J)),I)) ).

% not_add_less1
tff(fact_1281_not__add__less2,axiom,
    ! [J: 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),J),I)),I)) ).

% not_add_less2
tff(fact_1282_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => 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),J),K))) ) ).

% add_less_mono1
tff(fact_1283_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_1284_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_1285_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: 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),M),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),M),N)) ) ) ).

% less_add_eq_less
tff(fact_1286_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_1287_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => 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),M),J))) ) ).

% trans_le_add2
tff(fact_1288_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => 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),J),M))) ) ).

% trans_le_add1
tff(fact_1289_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => 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),J),K))) ) ).

% add_le_mono1
tff(fact_1290_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( 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),J),L))) ) ) ).

% add_le_mono
tff(fact_1291_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_1292_add__leD2,axiom,
    ! [M: 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),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% add_leD2
tff(fact_1293_add__leD1,axiom,
    ! [M: 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),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% add_leD1
tff(fact_1294_le__add2,axiom,
    ! [N: nat,M: 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),M),N))) ).

% le_add2
tff(fact_1295_le__add1,axiom,
    ! [N: nat,M: 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),M))) ).

% le_add1
tff(fact_1296_add__leE,axiom,
    ! [M: 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),M),K)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% add_leE
tff(fact_1297_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: 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),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% Nat.diff_cancel
tff(fact_1298_diff__cancel2,axiom,
    ! [M: 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),M),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),M),N) ).

% diff_cancel2
tff(fact_1299_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_1300_diff__add__inverse2,axiom,
    ! [M: 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),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1301_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1302_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_1303_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_1304_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y4: 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),Y4),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y4 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1305_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y4: 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)),Y4))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y4 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1306_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1307_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1308_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: 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),A2))
           => 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),A2),C2))) ) ) ) ).

% add_increasing2
tff(fact_1309_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_1310_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),C2))) ) ) ) ).

% add_increasing
tff(fact_1311_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_1312_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & 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),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1313_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & 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),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1314_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_le_less_mono
tff(fact_1315_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_less_le_mono
tff(fact_1316_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_1317_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_1318_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% add_pos_pos
tff(fact_1319_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1320_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),C2))) ) ) ) ).

% pos_add_strict
tff(fact_1321_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_1322_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A,J: 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),J),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),J),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)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1323_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: 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),A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_1324_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_1325_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),A2) = B2 ) ) ) ).

% diff_add
tff(fact_1326_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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)),A2))) ) ) ).

% le_add_diff
tff(fact_1327_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2)))
          <=> 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),A2)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1328_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1329_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1330_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1331_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1332_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1333_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1334_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1335_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))) ) ).

% less_add_one
tff(fact_1336_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_1337_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_1338_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_1339_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: 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),A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_1340_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1341_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_1342_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: 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),A2),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),A2),B2)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_1343_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: 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),A2),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),A2)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_1344_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y4: 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),Y4),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) ) ).

% square_diff_square_factored
tff(fact_1345_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y4: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y4),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1346_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),Z2))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))) ) ) ) ).

% minf(10)
tff(fact_1347_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),Z2))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))) ) ) ) ).

% minf(9)
tff(fact_1348_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z2),X4))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))) ) ) ) ).

% pinf(10)
tff(fact_1349_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z2),X4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))) ) ) ) ).

% pinf(9)
tff(fact_1350_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1351_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1352_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_add
tff(fact_1353_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1354_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1355_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1356_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1357_less__add__Suc2,axiom,
    ! [I: nat,M: 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),M),I)))) ).

% less_add_Suc2
tff(fact_1358_less__add__Suc1,axiom,
    ! [I: nat,M: 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),M)))) ).

% less_add_Suc1
tff(fact_1359_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ~ ! [Q4: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)) ) ).

% less_natE
tff(fact_1360_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ? [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) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1361_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,M4)),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,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_1362_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1363_diff__add__0,axiom,
    ! [N: nat,M: 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),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1364_less__diff__conv,axiom,
    ! [I: nat,J: 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),J),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)),J)) ) ).

% less_diff_conv
tff(fact_1365_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1366_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_1367_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1368_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_1369_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1370_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1371_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1372_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( 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),J),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)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1373_le__diff__conv,axiom,
    ! [J: 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),J),K)),I))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).

% le_diff_conv
tff(fact_1374_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X: nat,Y4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),D2) ) )
         => ? [X3: nat,Y: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A2))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1375_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X3: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A2))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_1376_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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),Y4),E))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% field_le_epsilon
tff(fact_1377_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_1378_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_1379_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_1380_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_1381_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_1382_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_1383_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y4: 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),Y4),Y4)))) ) ).

% sum_squares_ge_zero
tff(fact_1384_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y4: 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),Y4),Y4))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1385_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y4: 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),Y4),Y4))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1386_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y4: 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),Y4),Y4))))
        <=> ( ( X != zero_zero(A) )
            | ( Y4 != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1387_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_1388_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_1389_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: 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),A2),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),A2),B2)),E2)),C2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1390_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: 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),A2),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),A2)),E2)),D2))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1391_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1392_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1393_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y4: A,Z: A,X: A,W: A] :
          ( ( Y4 != 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),Y4)),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),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1394_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y4: A,X: A,Z: A] :
          ( ( Y4 != 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),Y4)),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),Y4))),Y4) ) ) ) ).

% add_frac_num
tff(fact_1395_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y4: A,Z: A,X: A] :
          ( ( Y4 != 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),Y4)) = 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),Y4))),Y4) ) ) ) ).

% add_num_frac
tff(fact_1396_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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),Y4),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)),Y4)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1397_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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)),Y4) = 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),Y4),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1398_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: 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),A2),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),A2),B2)),E2)),C2)),D2)) ) ) ).

% less_add_iff1
tff(fact_1399_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: 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),A2),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),A2)),E2)),D2))) ) ) ).

% less_add_iff2
tff(fact_1400_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1401_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1402_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: 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),M))))
     => ( 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)),M))
         => 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))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1403_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).

% less_half_sum
tff(fact_1404_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).

% gt_half_sum
tff(fact_1405_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_1406_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X5: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X5)))
        <=> ? [X5: 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),X5),zero_zero(A))))
              & pp(aa(A,bool,P,X5)) ) ) ) ).

% unity_coeff_ex
tff(fact_1407_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D4))
         => ! [X4: 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),X4),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),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T2))) ) ) ) ).

% inf_period(3)
tff(fact_1408_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D4))
         => ! [X4: 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),X4),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),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T2))) ) ) ) ).

% inf_period(4)
tff(fact_1409_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_1410_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1411_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_1412_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).

% less_diff_conv2
tff(fact_1413_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1414_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1415_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( 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)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1416_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),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)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,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),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1417_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1418_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1419_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X3: nat,Y: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A2))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1420_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A2: A,Y4: A,U: A,V: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),A2))
           => ( 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)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = 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),V),Y4))),A2)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1421_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_1422_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_1423_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_1424_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_1425_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: 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))),A2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),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),A2),B2))) ) ) ) ).

% odd_even_add
tff(fact_1426_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_1427_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_1428_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1429_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( 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)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1430_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),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)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,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),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1431_dividend__less__times__div,axiom,
    ! [N: nat,M: 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),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))))) ) ).

% dividend_less_times_div
tff(fact_1432_dividend__less__div__times,axiom,
    ! [N: nat,M: 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),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1433_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J3) )
               => pp(aa(nat,bool,P,I2)) ) ) ) ) ) ).

% split_div
tff(fact_1434_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1435_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),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),R2),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_1436_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A2: A,Y4: A,U: A,V: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),A2))
           => ( 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)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = 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),V),Y4))),A2)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1437_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),S2))
             => 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),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S2))),V)) ) ) ) ) ).

% scaling_mono
tff(fact_1438_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,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),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% field_less_half_sum
tff(fact_1439_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1440_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1441_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_1442_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_1443_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_1444_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_1445_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y4 = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1446_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_1447_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y4 != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1448_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_1449_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y4),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)),Y4)) ) ).

% power2_sum
tff(fact_1450_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_1451_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_1452_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_1453_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: 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,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y4)),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_squares_bound
tff(fact_1454_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)),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),Y4),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)),Y4)) ) ).

% power2_diff
tff(fact_1455_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_1456_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_1457_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y4: 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),Y4) )
         => ( 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)),Y4))
             => 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),Y4)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1458_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y4: 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),Y4),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_1459_low__inv,axiom,
    ! [X: nat,N: nat,Y4: 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),Y4),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_1460_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,bool))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [A4: real,B3: real,C4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),P,B3),C4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A4),B3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B3),C4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),C4)) ) ) ) )
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
               => ? [D6: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                    & ! [A4: real,B3: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A4),X3))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B3))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B3),A4)),D6)) )
                       => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B3)) ) ) ) )
         => pp(aa(real,bool,aa(real,fun(real,bool),P,A2),B2)) ) ) ) ).

% Bolzano
tff(fact_1461_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y4: 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),Y4),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1462_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y4: 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),Y4)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1463_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: 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)),K5))
           => ( 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))),K5))
             => 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),K5),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_1464_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_1465_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_1466_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ? [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X3))) ) )
       => ? [Y: A] :
            ( pp(aa(A,bool,P,Y))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1467_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(6)
tff(fact_1468_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_1469_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = aa(num,num,bit0,one2) ).

% semiring_norm(2)
tff(fact_1470_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_1471_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(7)
tff(fact_1472_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(9)
tff(fact_1473_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_1474_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_1475_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_1476_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_add_numeral2
tff(fact_1477_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_add_numeral
tff(fact_1478_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_1479_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit0,N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_1480_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% semiring_norm(4)
tff(fact_1481_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_1482_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_1483_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one2)) ).

% semiring_norm(10)
tff(fact_1484_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).

% semiring_norm(16)
tff(fact_1485_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_1486_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_1487_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_1488_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_1489_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_1490_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( N = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_1491_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_1492_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_1493_int__plus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_1494_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_1495_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z) ).

% zadd_int_left
tff(fact_1496_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_1497_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_1498_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_1499_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z))
    <=> ? [N4: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N4)) ) ).

% zle_iff_zadd
tff(fact_1500_zdvd__period,axiom,
    ! [A2: int,D2: int,X: int,T2: int,C2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A2),D2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A2),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),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),T2))) ) ) ).

% zdvd_period
tff(fact_1501_zdvd__reduce,axiom,
    ! [K: int,N: int,M: 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),M))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N)) ) ).

% zdvd_reduce
tff(fact_1502_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y4: extended_enat,X: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),Z),Y4))
     => ( 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),Y4),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),Y4)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1503_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_1504_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_1505_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
    <=> ? [N4: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_1506_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_1507_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_1508_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_1509_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_1510_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X3)),aa(A,nat,M,Y3))) ) ) ) ).

% ex_has_least_nat
tff(fact_1511_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N4))),K6)) )
        <=> ? [N6: nat] :
            ! [N4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% lemma_NBseq_def2
tff(fact_1512_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N4))),K6)) )
        <=> ? [N6: nat] :
            ! [N4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% lemma_NBseq_def
tff(fact_1513_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_1514_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_1515_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> 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),M))) ) ).

% nat_less_real_le
tff(fact_1516_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
    <=> 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),M)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_1517_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_1518_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: 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),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( 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),Q2),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1519_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: 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),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2)))
     => ( pp(aa(int,bool,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),R2),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1520_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),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),B5),Q5)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),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)),B5))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B5),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1521_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),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),B5),Q5)),R4)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B5))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B5),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q2),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1522_q__pos__lemma,axiom,
    ! [B5: int,Q5: 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),B5),Q5)),R4)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B5))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_1523_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))
     => ( ! [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),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X4: int] :
              ( pp(aa(int,bool,P,X4))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1524_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_1525_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_1526_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1527_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1528_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_1529_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_1530_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))
     => ( ! [Y: A] :
            ( pp(aa(A,bool,P,Y))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P,X3))
            & ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1531_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : 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(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1532_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: 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),M))))
     => ( 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)),M))
         => 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_1533_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int)))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_1534_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1535_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_1536_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y4: 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),Y4),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ) ).

% mult_less_iff1
tff(fact_1537_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),A2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_second
tff(fact_1538_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_first
tff(fact_1539_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_1540_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_1541_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_1542_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_1543_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_1544_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_1545_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_1546_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_1547_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W)) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_numeral
tff(fact_1548_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% norm_minus_commute
tff(fact_1549_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S2))
           => 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),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S2))) ) ) ) ).

% norm_triangle_mono
tff(fact_1550_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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),plus_plus(A),X),Y4))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y4)))) ) ).

% norm_triangle_ineq
tff(fact_1551_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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,Y4))),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),plus_plus(A),X),Y4))),E2)) ) ) ).

% norm_triangle_le
tff(fact_1552_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: 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),plus_plus(A),A2),B2))),C2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C2))) ) ) ).

% norm_add_leD
tff(fact_1553_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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),Y4))),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),Y4),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_1554_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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,Y4)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4))))) ) ).

% norm_triangle_sub
tff(fact_1555_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: 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),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))) ) ).

% norm_triangle_ineq4
tff(fact_1556_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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),Y4))),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),Y4),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_1557_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: 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,Y4))),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),Y4))),E2)) ) ) ).

% norm_triangle_le_diff
tff(fact_1558_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real))) ) ).

% norm_not_less_zero
tff(fact_1559_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X))) ) ).

% norm_ge_zero
tff(fact_1560_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y4: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y4)) ) ).

% norm_mult
tff(fact_1561_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_1562_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: 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),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))))) ) ).

% norm_diff_triangle_ineq
tff(fact_1563_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_1564_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_1565_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R2: real,Y4: A,S2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y4)),S2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S2))) ) ) ) ).

% norm_mult_less
tff(fact_1566_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y4: 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),times_times(A),X),Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y4)))) ) ).

% norm_mult_ineq
tff(fact_1567_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,X)),N))) ) ).

% norm_power_ineq
tff(fact_1568_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: 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,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))) ) ).

% norm_diff_ineq
tff(fact_1569_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: 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,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% norm_triangle_ineq2
tff(fact_1570_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_1571_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_1572_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: 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),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W))))) ) ) ) ).

% norm_power_diff
tff(fact_1573_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1574_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_1575_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1576_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_1577_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_1578_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_1579_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1580_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,modulo_modulo(A,A2,B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mod_trivial
tff(fact_1581_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_1582_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_1583_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_1584_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1585_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self2
tff(fact_1586_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self1
tff(fact_1587_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_1588_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_1589_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_1590_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_1591_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_1592_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_1593_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_1594_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = M ) ) ).

% mod_less
tff(fact_1595_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_1596_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1597_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1598_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1599_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1600_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1601_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1602_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_1603_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_1604_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_1605_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_1606_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1607_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_1608_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_1609_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_1610_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_1611_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1612_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_1613_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_1614_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self1
tff(fact_1615_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self2
tff(fact_1616_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self3
tff(fact_1617_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self4
tff(fact_1618_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_1619_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_1620_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% even_mod_2_iff
tff(fact_1621_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P3: bool] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(bool,A,zero_neq_one_of_bool(A),P3)))
        <=> pp(P3) ) ) ).

% odd_of_bool_self
tff(fact_1622_mod2__Suc__Suc,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% mod2_Suc_Suc
tff(fact_1623_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_1624_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1625_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1626_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_1627_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_1628_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1629_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1630_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_1631_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1632_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_1633_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_1634_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_1635_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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))),A2))
         => ( 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)),A2),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,A2,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_1636_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: bool,Q2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P3) = aa(bool,A,zero_neq_one_of_bool(A),Q2) )
        <=> ( pp(P3)
          <=> pp(Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_1637_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mod
tff(fact_1638_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_1639_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1640_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1641_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A2,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1642_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1643_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B5: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A5),B5),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1644_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_eq
tff(fact_1645_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_right_eq
tff(fact_1646_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_left_eq
tff(fact_1647_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B5: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B5),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1648_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_eq
tff(fact_1649_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1650_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1651_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B5: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A5),B5),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1652_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_1653_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_1654_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),modulo_modulo(A,A2,B2)),N),B2) = modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N),B2) ) ).

% power_mod
tff(fact_1655_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),modulo_modulo(A,M,N))) ) ) ) ).

% dvd_mod
tff(fact_1656_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_1657_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: 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,A2,B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2)) ) ) ) ).

% dvd_mod_iff
tff(fact_1658_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),modulo_modulo(A,A2,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),A2)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1659_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M,N))),N) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),N) ).

% mod_Suc_Suc_eq
tff(fact_1660_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M,N)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% mod_Suc_eq
tff(fact_1661_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),M)) ).

% mod_less_eq_dividend
tff(fact_1662_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_1663_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1664_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: 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,A2,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1665_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_1666_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1667_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1668_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_1669_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_1670_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1671_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1672_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ).

% mod_0_imp_dvd
tff(fact_1673_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_add1_eq
tff(fact_1674_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_1675_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P3: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P3)))
        <=> ~ ( ( pp(P3)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P3)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1676_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P3: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P3)))
        <=> ( ( pp(P3)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P3)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_1677_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: bool] :
          ( ( pp(P3)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P3) = one_one(A) ) )
          & ( ~ pp(P3)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P3) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_1678_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ) ).

% mod_eq_dvd_iff
tff(fact_1679_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: 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),A2),modulo_modulo(A,A2,B2)))) ) ).

% dvd_minus_mod
tff(fact_1680_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = aa(nat,nat,suc,modulo_modulo(nat,M,N)) ) ) ) ).

% mod_Suc
tff(fact_1681_mod__induct,axiom,
    ! [P: fun(nat,bool),N: nat,P3: nat,M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P3))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),P3))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P3))) ) )
           => pp(aa(nat,bool,P,M)) ) ) ) ) ).

% mod_induct
tff(fact_1682_mod__less__divisor,axiom,
    ! [N: nat,M: 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,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_1683_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M4),zero_zero(nat)))
     => ( ! [M4: 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,M4,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M4),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1684_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_1685_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( modulo_modulo(nat,M,D2) = zero_zero(nat) )
     => ? [Q4: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q4) ) ).

% mod_eq_0D
tff(fact_1686_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = M ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ) ).

% mod_if
tff(fact_1687_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% mod_geq
tff(fact_1688_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% le_mod_geq
tff(fact_1689_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y4: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y4,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1690_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_1691_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1692_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1693_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),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),M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ).

% exp_mod_exp
tff(fact_1694_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: 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,A2,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1695_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1696_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: 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,Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1697_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_1698_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_1699_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_1700_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_1701_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_1702_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_1703_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_1704_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_1705_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_1706_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1707_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1708_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1709_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1710_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1711_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1712_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1713_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_1714_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_1715_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1716_mod__le__divisor,axiom,
    ! [N: nat,M: 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,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_1717_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1718_div__less__mono,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A3,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ) ) ).

% div_less_mono
tff(fact_1719_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ~ ! [S3: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_1720_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ~ ! [S3: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_1721_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y4: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y4,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),X))
       => ? [Q4: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1722_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_1723_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1724_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N),Q2))),modulo_modulo(nat,M,N)) ).

% mod_mult2_eq
tff(fact_1725_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] : A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),N)),N)),modulo_modulo(nat,A3,N)) ).

% div_mod_decomp
tff(fact_1726_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)) ).

% modulo_nat_def
tff(fact_1727_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_1728_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_1729_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_1730_even__even__mod__4__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))),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),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,aa(num,num,bit0,one2)))))) ) ).

% even_even_mod_4_iff
tff(fact_1731_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N)))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_1732_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I2)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1733_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R2)))
         => ( modulo_modulo(nat,M,N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1734_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_1735_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_1736_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_1737_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_1738_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).

% pochhammer_product'
tff(fact_1739_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_1740_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_1741_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_1742_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_1743_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1744_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1745_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_1746_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_1747_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1748_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),N) ).

% Suc_mod_eq_add3_mod
tff(fact_1749_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1750_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% pochhammer_product
tff(fact_1751_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_1752_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_1753_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_1754_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_1755_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: 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,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1756_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1757_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1758_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_1759_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_1760_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
             => pp(aa(A,bool,P,A4)) )
         => ( ! [A4: A,B3: bool] :
                ( pp(aa(A,bool,P,A4))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
                 => pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4)))) ) )
           => pp(aa(A,bool,P,A2)) ) ) ) ).

% bits_induct
tff(fact_1761_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B4))
     => ( 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),A3),N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B4,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ).

% verit_le_mono_div
tff(fact_1762_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: 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,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1763_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),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,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1764_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
         => ( 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))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1765_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1766_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_1767_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_1768_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_1769_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_1770_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),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))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

% exp_div_exp_eq
tff(fact_1771_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_1772_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1773_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_1774_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_1775_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_1776_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_1777_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_1778_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_1779_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_1780_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_1781_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_1782_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_1783_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_1784_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_1785_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_1786_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M,K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_1787_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_1788_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_1789_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( modulo_modulo(int,M,D2) = zero_zero(int) )
     => ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0D
tff(fact_1790_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( modulo_modulo(int,M,D2) = zero_zero(int) )
    <=> ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0_iff
tff(fact_1791_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A2,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_1792_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_1793_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_1794_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_1795_pos__mod__conj,axiom,
    ! [B2: int,A2: 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,A2,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A2,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_1796_neg__mod__conj,axiom,
    ! [B2: int,A2: 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,A2,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A2,B2))) ) ) ).

% neg_mod_conj
tff(fact_1797_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( ( modulo_modulo(int,A3,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,N) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1798_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] : A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N)),N)),modulo_modulo(int,A3,N)) ).

% div_mod_decomp_int
tff(fact_1799_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_1800_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_1801_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_1802_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_1803_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1804_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1805_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C2))
     => ( modulo_modulo(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1806_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B4))
     => ( 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),A3),N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B4,N)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_1807_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_1808_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_1809_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_1810_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int)))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_1811_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_1812_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_1813_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_1814_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_1815_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),M)) ).

% choose_mult_lemma
tff(fact_1816_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% binomial_le_pow
tff(fact_1817_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_1818_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_1819_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% choose_mult
tff(fact_1820_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_1821_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_1822_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_1823_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_1824_binomial__le__pow2,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% binomial_le_pow2
tff(fact_1825_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_1826_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_1827_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,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,M,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_1828_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_1829_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1830__C10_C,axiom,
    ! [L: nat] : foldr(real,real,plus_plus(real),replicate(real,L,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,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)))))),zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),L)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,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)))))) ).

% "10"
tff(fact_1831_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1832_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_1833_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : size_num(aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_1834_foldr0,axiom,
    ! [Xs: list(real),C2: real,D2: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D2)),C2) ).

% foldr0
tff(fact_1835_ln__less__cancel__iff,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( 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),Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_1836_ln__inj__iff,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y4) )
        <=> ( X = Y4 ) ) ) ) ).

% ln_inj_iff
tff(fact_1837_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_1838_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_1839_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat,Y4: A] :
      ( ( replicate(A,M,X) = replicate(A,N,Y4) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( X = Y4 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1840_ln__le__cancel__iff,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( 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),Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1841_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_1842_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_1843_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_1844_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1845_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_1846_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_1847_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_1848_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_1849_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_1850_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_1851_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_1852_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% fact_2
tff(fact_1853_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_1854_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_1855_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_1856_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_1857_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_1858_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_1859_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_1860_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% signed_take_bit_diff
tff(fact_1861_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_1862_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_1863_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_1864_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_1865_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_1866_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_1867_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_1868_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_1869_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% fact_diff_Suc
tff(fact_1870_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_1871_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_1872_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_1873_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_1874_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_1875_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_1876_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_1877_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_1878_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] : 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,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_1879_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_1880_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_1881_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_1882_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_1883_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_1884_ln__mult,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y4)) ) ) ) ).

% ln_mult
tff(fact_1885_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_1886_ln__div,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y4)) ) ) ) ).

% ln_div
tff(fact_1887_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),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),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_1888_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_1889_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_1890_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_1891_ln__diff__le,axiom,
    ! [X: real,Y4: 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)),Y4))
       => 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),Y4))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4)),Y4))) ) ) ).

% ln_diff_le
tff(fact_1892_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_1893_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_1894_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_1895_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1896_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: 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_ri4674362597316999326ke_bit(A,M),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% even_signed_take_bit_iff
tff(fact_1897_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_1898_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( ( M = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_1899_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_1900_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,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_1901_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),aa(int,int,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_1902_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_1903_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_1904_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_1905_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y4: A,X: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y4)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)) )
        <=> ( ( W = X )
            | ( Y4 = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1906_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1907_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),aa(int,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_1908_eq__diff__eq_H,axiom,
    ! [X: real,Y4: real,Z: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),minus_minus(real),Y4),Z) )
    <=> ( Y4 = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Z) ) ) ).

% eq_diff_eq'
tff(fact_1909_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_1910_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(aa(num,num,bit0,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_1911_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_1912_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_1913_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_1914_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,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_1915_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_1916_map__cnt,axiom,
    foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,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,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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),va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),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,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% map_cnt
tff(fact_1917_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_1918_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_1919_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1920_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_1921_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B4: 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)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3)) ) ).

% Compl_subset_Compl_iff
tff(fact_1922_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => 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)),B4)),aa(set(A),set(A),uminus_uminus(set(A)),A3))) ) ).

% Compl_anti_mono
tff(fact_1923_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_1924_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_1925_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: 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),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% neg_le_iff_le
tff(fact_1926_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1927_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1928_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1929_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1930_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_1931_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: 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),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% neg_less_iff_less
tff(fact_1932_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_1933_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_1934_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_1935_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_1936_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_1937_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_1938_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_1939_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% minus_diff_eq
tff(fact_1940_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% div_minus_minus
tff(fact_1941_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_1942_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1943_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_1944_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_1945_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4)) ) ) ).

% minus_dvd_iff
tff(fact_1946_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4)) ) ) ).

% dvd_minus_iff
tff(fact_1947_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_1948_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_1949_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_1950_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_1951_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_1952_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_1953_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_1954_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)) ) ).

% mod_minus_minus
tff(fact_1955_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),aa(A,A,abs_abs(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).

% dvd_abs_iff
tff(fact_1956_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,abs_abs(A),M)),K))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).

% abs_dvd_iff
tff(fact_1957_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_1958_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_1959_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_1960_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_1961_tanh__real__le__iff,axiom,
    ! [X: real,Y4: 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),Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ).

% tanh_real_le_iff
tff(fact_1962_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_less_eq_nonneg
tff(fact_1963_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_1964_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_le_0_iff_le
tff(fact_1965_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_1966_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_1967_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_pos
tff(fact_1968_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_1969_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_0_iff_less
tff(fact_1970_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_1971_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_1972_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_1973_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_1974_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1975_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_1976_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_1977_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1978_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% abs_le_self_iff
tff(fact_1979_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_1980_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_1981_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_1982_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1983_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A2) ) ).

% div_minus1_right
tff(fact_1984_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_1985_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_1986_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_1987_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_1988_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_1989_real__add__minus__iff,axiom,
    ! [X: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( X = A2 ) ) ).

% real_add_minus_iff
tff(fact_1990_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_1991_negative__zle,axiom,
    ! [N: nat,M: 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),M))) ).

% negative_zle
tff(fact_1992_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_1993_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_1994_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_1995_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_1996_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_1997_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_1998_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_1999_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_2000_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_2001_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_2002_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_2003_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2004_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: 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),A2),aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2005_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2006_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_2007_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_2008_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2009_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_2010_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y4) ) ).

% semiring_norm(168)
tff(fact_2011_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2012_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2013_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Y4) ) ).

% semiring_norm(172)
tff(fact_2014_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y4) ) ).

% semiring_norm(171)
tff(fact_2015_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y4) ) ).

% semiring_norm(170)
tff(fact_2016_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2017_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2018_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2019_negative__zless,axiom,
    ! [N: nat,M: 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),M))) ).

% negative_zless
tff(fact_2020_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),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),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_2021_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),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),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_2022_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2023_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: 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)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2024_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2025_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A2 )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2026_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2027_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2028_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: 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)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2029_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2030_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_minus
tff(fact_2031_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),N)))
        <=> ( ( A2 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2032_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% abs_power2
tff(fact_2033_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_abs
tff(fact_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2041_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: 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),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ).

% power_minus_odd
tff(fact_2042_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: 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),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2043_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: 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)))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2044_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_2045_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2046_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_2047_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_2048_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_2049_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_2050_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_2051_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2052_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2053_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_2054_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_2055_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y4) )
        <=> ( ( X = Y4 )
            | ( X = aa(A,A,uminus_uminus(A),Y4) ) ) ) ) ).

% abs_eq_iff
tff(fact_2056_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2057_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_minus_self
tff(fact_2058_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2059_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ).

% abs_le_D2
tff(fact_2060_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)) ) ) ) ).

% abs_leI
tff(fact_2061_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2062_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2063_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2064_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_2065_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X4) = aa(A,A,uminus_uminus(A),X4) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X4) = X4 ) ) ) ) ).

% abs_if_raw
tff(fact_2066_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_2067_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ) ).

% abs_if
tff(fact_2068_abs__real__def,axiom,
    ! [A2: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = aa(real,real,uminus_uminus(real),A2) ) )
      & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = A2 ) ) ) ).

% abs_real_def
tff(fact_2069_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% abs_le_D1
tff(fact_2070_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_self
tff(fact_2071_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2072_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2073_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2074_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% abs_minus_commute
tff(fact_2075_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_2076_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2))) ) ) ).

% le_minus_iff
tff(fact_2077_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_le_iff
tff(fact_2078_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2))) ) ) ).

% le_imp_neg_le
tff(fact_2079_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2))) ) ) ).

% less_minus_iff
tff(fact_2080_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_less_iff
tff(fact_2081_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2082_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_2083_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_2084_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A2 = B2 )
            | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2085_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_2086_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_2087_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_2088_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% is_num_normalize(8)
tff(fact_2089_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_2090_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_2091_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ).

% minus_diff_minus
tff(fact_2092_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% div_minus_right
tff(fact_2093_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_2094_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% minus_divide_divide
tff(fact_2095_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_divide_left
tff(fact_2096_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% mod_minus_eq
tff(fact_2097_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A5: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A5,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A5),B2) ) ) ) ).

% mod_minus_cong
tff(fact_2098_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2)) ) ).

% mod_minus_right
tff(fact_2099_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_2100_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_2101_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_2102_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_2103_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_zero
tff(fact_2104_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2105_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_2106_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_2107_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),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),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))) ) ) ) ).

% abs_mult_less
tff(fact_2108_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_2109_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_2110_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2111_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2112_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_2113_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2114_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_2115_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2116_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_2117_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_2118_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_2119_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2120_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2121_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2122_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2123_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_2124_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_2125_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_2126_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_2127_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_2128_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2129_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2130_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_2131_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_2132_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_2133_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2134_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_2135_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2136_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2137_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2138_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_2139_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_2140_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_2141_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2142_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2143_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_2144_not__int__zless__negative,axiom,
    ! [N: nat,M: 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),M)))) ).

% not_int_zless_negative
tff(fact_2145_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_2146_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_2147_minus__real__def,axiom,
    ! [X: real,Y4: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y4)) ).

% minus_real_def
tff(fact_2148_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_2149_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2150_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2151_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),N))) ) ).

% zero_le_power_abs
tff(fact_2152_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y4))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y4) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4)) ) ) ) ).

% abs_div_pos
tff(fact_2153_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: 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),A2))),R2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),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),A2),R2))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2154_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_2155_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2156_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: 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),A2))),R2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),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),A2),R2))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2157_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_2158_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_2159_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_2160_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_2161_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_2162_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_2163_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_2164_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_2165_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_2166_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))) ) ).

% neg_one_le_numeral
tff(fact_2167_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_2168_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_2169_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_2170_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_2171_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M))) ) ).

% neg_one_less_numeral
tff(fact_2172_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_2173_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_2174_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: 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),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2175_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2176_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2177_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2178_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2179_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_2180_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_2181_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2182_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_2183_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_minus
tff(fact_2184_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_2185_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [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))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2186_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y4: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),Y4)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) ) ).

% norm_uminus_minus
tff(fact_2187_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_2188_sin__bound__lemma,axiom,
    ! [X: real,Y4: real,U: real,V: real] :
      ( ( X = Y4 )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V))
       => 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)),Y4))),V)) ) ) ).

% sin_bound_lemma
tff(fact_2189_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != 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))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_2190_real__0__less__add__iff,axiom,
    ! [X: real,Y4: 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),Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y4)) ) ).

% real_0_less_add_iff
tff(fact_2191_real__add__less__0__iff,axiom,
    ! [X: real,Y4: 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),Y4)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_less_0_iff
tff(fact_2192_real__add__le__0__iff,axiom,
    ! [X: real,Y4: 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),Y4)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_le_0_iff
tff(fact_2193_real__0__le__add__iff,axiom,
    ! [X: real,Y4: 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),Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),Y4)) ) ).

% real_0_le_add_iff
tff(fact_2194_int__zle__neg,axiom,
    ! [N: nat,M: 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),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2195_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_2196_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_2197_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A2,B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_2198_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A2,B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_2199_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_2200_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,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),A2),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),A2),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),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2201_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: 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))),A2))
        <=> ( ( 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),A2),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),A2),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)),A2)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2202_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2203_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2204_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2205_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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))),A2))
          <=> 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),A2),C2))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2206_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_2207_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_2208_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2209_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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))),Y4) = 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),Y4),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2210_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = 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),A2),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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2211_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2212_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y4: 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))),Y4) = 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),Y4),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2213_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: 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),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% even_minus
tff(fact_2214_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [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))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_2215_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
        <=> ( ( X = Y4 )
            | ( X = aa(A,A,uminus_uminus(A),Y4) ) ) ) ) ).

% power2_eq_iff
tff(fact_2216_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: 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,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% norm_triangle_ineq3
tff(fact_2217_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_2218_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B4))
     => ( 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),B4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_2227_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_2228_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y4)))
        <=> 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2229_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_2230_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: 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,abs_abs(A),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% power_even_abs
tff(fact_2231_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,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),A2),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),A2),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),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2232_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: 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))),A2))
        <=> ( ( 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),A2),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),A2),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)),A2)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2233_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2234_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2235_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: 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),A2),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),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2236_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: 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))),A2))
          <=> 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),A2),C2))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2237_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_2238_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_2239_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( ( A2 = one_one(A) )
            | ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2240_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: 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),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),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),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ) ).

% uminus_power_if
tff(fact_2241_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_2242_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_2243_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_2244_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_2245_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),aa(int,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_2246_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))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2247_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),aa(int,int,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_2248_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_2249_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_2250_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2251_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2252_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y4))
         => ( 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),Y4),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)),Y4)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2253_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_2254_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_2255_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: 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),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(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),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2256_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_2257_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_2258_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_2259_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_2260_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% minus_power_mult_self
tff(fact_2261_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_2262_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2263_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_2264_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,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_2265_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)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2266_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_2267_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_2268_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_2269_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_2270_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_2271_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_2272_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_2273_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)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2274_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_2275_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_2276_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,bool)),X: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),aa(nat,A,aa(A,fun(nat,A),power_power(A),X3),aa(num,nat,numeral_numeral(nat),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_2277_sin__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X4))
       => ( sin_coeff(X4) = 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))),X4))
       => ( sin_coeff(X4) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X4),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X4)) ) ) ) ).

% sin_coeff_def
tff(fact_2278_cos__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X4))
       => ( cos_coeff(X4) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X4)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X4))
       => ( cos_coeff(X4) = zero_zero(real) ) ) ) ).

% cos_coeff_def
tff(fact_2279_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: 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),Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ).

% compl_le_compl_iff
tff(fact_2280_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Y4: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X3))) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),Y4))) ) ).

% listsum_bound
tff(fact_2281_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2282_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_2283_ComplI,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A3))) ) ).

% ComplI
tff(fact_2284_Compl__iff,axiom,
    ! [A: $tType,C2: A,A3: 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)),A3)))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% Compl_iff
tff(fact_2285_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B4) )
    <=> ( A3 = B4 ) ) ).

% Compl_eq_Compl_iff
tff(fact_2286_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_2287_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_2288_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_2289_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y4: 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,Y4))))
    <=> ( ( X = Y4 )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_2290_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
          & pp(aa(A,bool,P,X5)) )
    <=> ( pp(aa(A,bool,P,A2))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_2291_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
         => pp(aa(A,bool,P,X5)) )
    <=> ( pp(aa(A,bool,P,A2))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_2292_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2293_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_2294_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_2295_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_2296_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2308_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2309_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2310_ComplD,axiom,
    ! [A: $tType,C2: A,A3: 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)),A3)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% ComplD
tff(fact_2311_double__complement,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = A3 ).

% double_complement
tff(fact_2312_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: 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)),B4))
    <=> ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4)) ) ) ).

% subset_code(1)
tff(fact_2313_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A2),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A2))
       => ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_2314_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2315_abs__div,axiom,
    ! [Y4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Y4),X))
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y4)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),X)),aa(int,int,abs_abs(int),Y4)) ) ) ).

% abs_div
tff(fact_2316_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_2317_sin__coeff__Suc,axiom,
    ! [N: nat] : sin_coeff(aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% sin_coeff_Suc
tff(fact_2318_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_2319_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_2320_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_2321_cos__coeff__Suc,axiom,
    ! [N: nat] : cos_coeff(aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% cos_coeff_Suc
tff(fact_2322_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_2323_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != 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),M),N)),M))
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2324_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_2325_even__abs__add__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),plus_plus(int),aa(int,int,abs_abs(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_abs_add_iff
tff(fact_2326_even__add__abs__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),plus_plus(int),K),aa(int,int,abs_abs(int),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_add_abs_iff
tff(fact_2327_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,M)),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),M),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_2328_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_2329_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_2330_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_2331_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y4)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_2332_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),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),Y4))) ) ) ).

% compl_le_swap1
tff(fact_2333_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y4)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y4)) ) ) ).

% compl_le_swap2
tff(fact_2334_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_2335_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_2336_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: 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),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2))) ).

% complex_mod_triangle_ineq2
tff(fact_2337_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_2338_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_2339_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_2340_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,N),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_concat_bit(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_2341_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: 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),M)),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),M)) ) )
      & ( ~ ( ( 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),M)),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,M,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),M)),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),M)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_2342_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_2343_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ).

% sgn_sgn
tff(fact_2344_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_2345_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_2346_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_2347_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_2348_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_divide
tff(fact_2349_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_2350_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_2351_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_2352_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_2353_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_2354_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_greater
tff(fact_2355_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_2356_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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)),aa(int,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_2363_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,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_2364_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2365_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_2366_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_2367_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_2368_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_2369_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2370_sgn__mult__dvd__iff,axiom,
    ! [R2: 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),R2)),L)),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_2371_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: 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),R2))),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_2372_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: 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),R2)),K)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_2373_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: 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),R2))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_2374_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2375_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_2376_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_2377_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y4) )
         => ( X != Y4 ) ) ) ).

% size_neq_size_imp_neq
tff(fact_2378_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_2379_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_2380_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_2381_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y4: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,sgn_sgn(A),Y4)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_2382_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_2383_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,sgn_sgn(A),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_2384_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_2385_arctan__monotone_H,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y4))) ) ).

% arctan_monotone'
tff(fact_2386_arctan__le__iff,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ).

% arctan_le_iff
tff(fact_2387_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_2388_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa(int,int,bit_concat_bit(N,K),aa(int,int,bit_concat_bit(M,L),R2)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(int,int,bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_2389_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2390_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_2391_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_2392_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,sgn_sgn(A),A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_2393_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_2394_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_2395_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_2396_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_2397_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_2398_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_2399_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_1_pos
tff(fact_2400_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_2401_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_2402_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_2403_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_2404_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_2405_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_2406_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_2407_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_arctan__add,axiom,
    ! [X: real,Y4: 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),Y4)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y4)) = 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),Y4)),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),Y4)))) ) ) ) ).

% arctan_add
tff(fact_2419_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: 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),M))),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),M))),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),M),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),M))),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),M),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),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_2420_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_2421_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_2422_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_2423_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_2424_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_2425_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_2426_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_2427_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_2428_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_2429_real__sqrt__eq__iff,axiom,
    ! [X: real,Y4: real] :
      ( ( aa(real,real,sqrt,X) = aa(real,real,sqrt,Y4) )
    <=> ( X = Y4 ) ) ).

% real_sqrt_eq_iff
tff(fact_2430_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sgn_sgn(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% sgn_le_0_iff
tff(fact_2431_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% zero_le_sgn_iff
tff(fact_2432_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_2433_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_2434_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_2435_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_2436_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_2437_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_2438_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_2439_real__sqrt__less__iff,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ).

% real_sqrt_less_iff
tff(fact_2440_real__sqrt__le__iff,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ).

% real_sqrt_le_iff
tff(fact_2441_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_2442_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_2443_nat__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),N)) = N ).

% nat_int
tff(fact_2444_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_and
tff(fact_2445_exp__le__cancel__iff,axiom,
    ! [X: real,Y4: 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),Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ).

% exp_le_cancel_iff
tff(fact_2446_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_concat_bit(N,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) ).

% concat_bit_of_zero_2
tff(fact_2447_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_2448_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_2449_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_2450_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_2451_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = A2 ) ).

% and.right_neutral
tff(fact_2452_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A2) = A2 ) ).

% and.left_neutral
tff(fact_2453_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_2454_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_2455_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% real_sqrt_lt_0_iff
tff(fact_2456_real__sqrt__gt__0__iff,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4)) ) ).

% real_sqrt_gt_0_iff
tff(fact_2457_real__sqrt__ge__0__iff,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4)) ) ).

% real_sqrt_ge_0_iff
tff(fact_2458_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% real_sqrt_le_0_iff
tff(fact_2459_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_2460_real__sqrt__gt__1__iff,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y4)) ) ).

% real_sqrt_gt_1_iff
tff(fact_2461_real__sqrt__ge__1__iff,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y4)) ) ).

% real_sqrt_ge_1_iff
tff(fact_2462_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_2463_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_2464_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_2465_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_2466_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A2)),aa(real,real,sqrt,A2)) = aa(real,real,abs_abs(real),A2) ).

% real_sqrt_mult_self
tff(fact_2467_real__sqrt__abs2,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),X)) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs2
tff(fact_2468_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_2469_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2470_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_2471_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_2472_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% real_sqrt_four
tff(fact_2473_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_2485_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_2486_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_2487_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_2488_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2489_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_2490_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y4))) = 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),Y4))) ) ).

% and_numerals(3)
tff(fact_2491_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_2492_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_2493_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_2494_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_2495_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_2496_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y4: int,X: num,N: nat] :
      ( ( aa(int,nat,nat2,Y4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y4 = 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_2497_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y4: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = aa(int,nat,nat2,Y4) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y4 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2498_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_2499_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_2500_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y4))) = 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),Y4))) ) ).

% and_numerals(4)
tff(fact_2501_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,bit0,Y4))) = 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),Y4))) ) ).

% and_numerals(6)
tff(fact_2502_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: 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_se2584673776208193580ke_bit(A,N),A2)))
        <=> ( ( 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))),A2)) ) ) ) ).

% even_take_bit_eq
tff(fact_2503_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_2504_real__sqrt__abs,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs
tff(fact_2505_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_2506_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_2507_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_2508_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_2509_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( 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)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X ) ) ).

% real_sqrt_pow2
tff(fact_2510_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y4: real,Xa: real,Ya: real] : aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_2511_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_2512_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: 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,A2)))
    <=> 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)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2513_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2514_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2515_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: 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,A2)))
    <=> 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)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2516_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% and_numerals(7)
tff(fact_2517_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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),M))),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_2518_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_2519_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,N)) ) ).

% take_bit_eq_mask
tff(fact_2520_nat__mask__eq,axiom,
    ! [N: nat] : aa(int,nat,nat2,bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(nat,N) ).

% nat_mask_eq
tff(fact_2521_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_2522_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_2523_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_2524_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_and_eq
tff(fact_2525_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)) ) ).

% take_bit_of_nat
tff(fact_2526_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,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_2527_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))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_2528_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_2529_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_2530_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2531_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_2532_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2533_real__sqrt__less__mono,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4))) ) ).

% real_sqrt_less_mono
tff(fact_2534_real__sqrt__le__mono,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4))) ) ).

% real_sqrt_le_mono
tff(fact_2535_real__sqrt__divide,axiom,
    ! [X: real,Y4: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4)) ).

% real_sqrt_divide
tff(fact_2536_real__sqrt__mult,axiom,
    ! [X: real,Y4: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4)) ).

% real_sqrt_mult
tff(fact_2537_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),K) ).

% real_sqrt_power
tff(fact_2538_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_2539_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_2540_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_2541_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,exp(A),A3)) ) ).

% exp_times_arg_commute
tff(fact_2542_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_2543_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% take_bit_diff
tff(fact_2544_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] : bit_concat_bit(N,aa(int,int,bit_se2584673776208193580ke_bit(int,N),B2)) = bit_concat_bit(N,B2) ).

% concat_bit_take_bit_eq
tff(fact_2545_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S2: int] :
      ( ( aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,bit_concat_bit(N,R2),S2) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),R2) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_2546_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_2547_real__sqrt__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),zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_2548_exp__total,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4))
     => ? [X3: real] : aa(real,real,exp(real),X3) = Y4 ) ).

% exp_total
tff(fact_2549_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_2550_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_2551_real__sqrt__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),zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_2552_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( aa(real,real,sqrt,X) = zero_zero(real) )
       => ( X = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_2553_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_2554_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_2555_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_2556_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_2557_AND__upper2_H,axiom,
    ! [Y4: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y4),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),Y4)),Z)) ) ) ).

% AND_upper2'
tff(fact_2558_AND__upper1_H,axiom,
    ! [Y4: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y4),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),Y4),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_2559_AND__upper2,axiom,
    ! [Y4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => 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),Y4)),Y4)) ) ).

% AND_upper2
tff(fact_2560_AND__upper1,axiom,
    ! [X: int,Y4: 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),Y4)),X)) ) ).

% AND_upper1
tff(fact_2561_AND__lower,axiom,
    ! [X: int,Y4: 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),Y4))) ) ).

% AND_lower
tff(fact_2562_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2563_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_2564_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),aa(int,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_2565_take__bit__nonnegative,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_se2584673776208193580ke_bit(int,N),K))) ).

% take_bit_nonnegative
tff(fact_2566_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,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_2567_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_2568_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2569_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_2570_nat__mono,axiom,
    ! [X: int,Y4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y4))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y4))) ) ).

% nat_mono
tff(fact_2571_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),A2) ) ).

% signed_take_bit_take_bit
tff(fact_2572_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: 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)),Z4))
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2573_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_12: nat] : pp(aa(nat,bool,P,X_12))
    <=> ! [X5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
         => pp(aa(nat,bool,P,aa(int,nat,nat2,X5))) ) ) ).

% all_nat
tff(fact_2574_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
    <=> ? [X5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
          & pp(aa(nat,bool,P,aa(int,nat,nat2,X5))) ) ) ).

% ex_nat
tff(fact_2575_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y4)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) ) ).

% mult_exp_exp
tff(fact_2576_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),X) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y4)) ) ) ) ).

% exp_add_commuting
tff(fact_2577_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y4)) ) ).

% exp_diff
tff(fact_2578_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_2579_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_2580_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_2581_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_2582_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),N) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_2583_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_2584_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_2585_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_2586_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y4: 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)),Y4))
       => 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),plus_plus(real),X),Y4))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y4)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_2587_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2588_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_2589_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y4),Y4))))) ).

% le_real_sqrt_sumsq
tff(fact_2590_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_2591_AND__upper1_H_H,axiom,
    ! [Y4: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y4),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),Y4),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_2592_AND__upper2_H_H,axiom,
    ! [Y4: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y4),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),Y4)),Z)) ) ) ).

% AND_upper2''
tff(fact_2593_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_2594_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_2595_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_2596_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_2597_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),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),M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_2598_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_2599_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_2600_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = 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_2601_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_2602_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_2603_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_2604_int__minus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = 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),M)))) ).

% int_minus
tff(fact_2605_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_2606_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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),bit_se5824344872417868541ns_and(A),A2),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_and_iff
tff(fact_2607_sqrt2__less__2,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% sqrt2_less_2
tff(fact_2608_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( aa(real,real,sgn_sgn(real),A2) = zero_zero(real) ) )
      & ( ( A2 != zero_zero(real) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( aa(real,real,sgn_sgn(real),A2) = one_one(real) ) )
          & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( aa(real,real,sgn_sgn(real),A2) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ) ).

% sgn_real_def
tff(fact_2609_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_2610_even__and__iff__int,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),bit_se5824344872417868541ns_and(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))),K))
        | pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L)) ) ) ).

% even_and_iff_int
tff(fact_2611_lemma__exp__total,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y4))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y4),one_one(real))))
          & ( aa(real,real,exp(real),X3) = Y4 ) ) ) ).

% lemma_exp_total
tff(fact_2612_ln__ge__iff,axiom,
    ! [X: real,Y4: 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),Y4),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),Y4)),X)) ) ) ).

% ln_ge_iff
tff(fact_2613_ln__x__over__x__mono,axiom,
    ! [X: real,Y4: 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),Y4))
       => 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),Y4)),Y4)),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_2614_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_2615_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_2616_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = 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),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_2617_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa(int,nat,nat2,W) = M )
    <=> ( ( 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),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_2618_split__nat,axiom,
    ! [P: fun(nat,bool),I: int] :
      ( pp(aa(nat,bool,P,aa(int,nat,nat2,I)))
    <=> ( ! [N4: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N4) )
           => pp(aa(nat,bool,P,N4)) )
        & ( 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_2619_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_2620_nat__add__distrib,axiom,
    ! [Z: int,Z4: 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)),Z4))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_2621_nat__mult__distrib,axiom,
    ! [Z: int,Z4: 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),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_2622_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),one_one(int))) ).

% Suc_as_int
tff(fact_2623_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),Z))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_2624_nat__diff__distrib_H,axiom,
    ! [X: int,Y4: 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)),Y4))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y4)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2625_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_2626_nat__div__distrib,axiom,
    ! [X: int,Y4: 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),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y4)) ) ) ).

% nat_div_distrib
tff(fact_2627_nat__div__distrib_H,axiom,
    ! [Y4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y4))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y4)) ) ) ).

% nat_div_distrib'
tff(fact_2628_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_2629_nat__mod__distrib,axiom,
    ! [X: int,Y4: 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)),Y4))
       => ( aa(int,nat,nat2,modulo_modulo(int,X,Y4)) = modulo_modulo(nat,aa(int,nat,nat2,X),aa(int,nat,nat2,Y4)) ) ) ) ).

% nat_mod_distrib
tff(fact_2630_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_2631_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2632_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_2633_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2634_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_eq_mod
tff(fact_2635_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_2636_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),one_one(A)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_2637_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2638_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% take_bit_nat_less_exp
tff(fact_2639_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2640_real__less__rsqrt,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,sqrt,Y4))) ) ).

% real_less_rsqrt
tff(fact_2641_real__le__rsqrt,axiom,
    ! [X: real,Y4: 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),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,Y4))) ) ).

% real_le_rsqrt
tff(fact_2642_sqrt__le__D,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),Y4))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% sqrt_le_D
tff(fact_2643_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_2644_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% take_bit_nat_def
tff(fact_2645_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_2646_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),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))
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_2647_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% take_bit_int_less_exp
tff(fact_2648_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_2649_nat__less__iff,axiom,
    ! [W: int,M: 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)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_2650_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = modulo_modulo(int,K,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% take_bit_int_def
tff(fact_2651_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: 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),Z4)) = 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),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2652_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_2653_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),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),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),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),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_2654_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_2655_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) )
        <=> pp(aa(A,bool,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)),A2)) ) ) ).

% take_bit_eq_0_iff
tff(fact_2656_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_2657_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M))
    <=> 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)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_2658_real__sqrt__unique,axiom,
    ! [Y4: real,X: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
       => ( aa(real,real,sqrt,X) = Y4 ) ) ) ).

% real_sqrt_unique
tff(fact_2659_real__le__lsqrt,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),Y4)) ) ) ) ).

% real_le_lsqrt
tff(fact_2660_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_2661_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = Y4 )
     => ( X = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_2662_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = X )
     => ( Y4 = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_2663_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : 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),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_2664_real__sqrt__sum__squares__ge2,axiom,
    ! [Y4: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_2665_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_2666_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_2667_sqrt__ge__absD,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y4)))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y4)) ) ).

% sqrt_ge_absD
tff(fact_2668_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_2669_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),aa(int,int,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_2670_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,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_2671_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_2672_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z)),M))
    <=> ( ( 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),M))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_2673_real__less__lsqrt,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),Y4)) ) ) ) ).

% real_less_lsqrt
tff(fact_2674_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y4: 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)),Y4))
       => 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),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y4))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_2675_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y4: real] : 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),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),Y4)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_2676_real__sqrt__ge__abs2,axiom,
    ! [Y4: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y4)),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_ge_abs2
tff(fact_2677_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_ge_abs1
tff(fact_2678_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_2679_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_2680_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_2681_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)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2682_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,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_2683_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_2684_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_2685_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,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)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2686_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> pp(aa(int,bool,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_2687_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_2688_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,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_2689_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2690_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2691_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2692_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_2693_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_2694_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y4: real,Xa: real,Ya: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_2695_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_2696_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y4: 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)),Y4))
       => 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),Y4))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_2697_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),aa(int,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_2698_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))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_2699_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_2700_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),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_2701_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2702_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = 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))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = 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_2703_num_Osize_I6_J,axiom,
    ! [X33: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2704_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2705_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_2706_cos__x__y__le__one,axiom,
    ! [X: real,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),one_one(real))) ).

% cos_x_y_le_one
tff(fact_2707_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y4: 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),Y4)),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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_2708_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_2709_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_2710_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_2711_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)))
       => ( aa(int,int,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_2712_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_2713_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_2714_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y4: 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),Y4),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)),Y4))
           => 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2715_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_2716_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2717_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2718_foldr__same__int,axiom,
    ! [Xs: list(nat),Y4: nat] :
      ( ! [X3: nat,Y: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(list(nat),set(nat),set2(nat),Xs)))
         => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),aa(list(nat),set(nat),set2(nat),Xs)))
           => ( X3 = Y ) ) )
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(list(nat),set(nat),set2(nat),Xs)))
           => ( X3 = Y4 ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y4) ) ) ) ).

% foldr_same_int
tff(fact_2719_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,log(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_2720_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,log(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,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_2721_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_2722_foldr__one,axiom,
    ! [D2: nat,Ys: list(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),D2),foldr(nat,nat,plus_plus(nat),Ys,D2))) ).

% foldr_one
tff(fact_2723_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_2724_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2725_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_2726_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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)),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y4)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2727_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2728_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2729_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),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_2730_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2731_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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)),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y4)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2732_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2733_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2734_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),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_2735_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( 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,log(A2),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_2736_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_2737_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_2738_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_2739_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_2740_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_2741_and__nat__numerals_I1_J,axiom,
    ! [Y4: 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,Y4))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2742_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_2743_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_2744_and__nat__numerals_I2_J,axiom,
    ! [Y4: 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,Y4))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2745_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_2746_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_2747_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_2748_num__induct,axiom,
    ! [P: fun(num,bool),X: num] :
      ( pp(aa(num,bool,P,one2))
     => ( ! [X3: num] :
            ( pp(aa(num,bool,P,X3))
           => pp(aa(num,bool,P,inc(X3))) )
       => pp(aa(num,bool,P,X)) ) ) ).

% num_induct
tff(fact_2749_add__inc,axiom,
    ! [X: num,Y4: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y4)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y4)) ).

% add_inc
tff(fact_2750_log__def,axiom,
    ! [A2: real,X: real] : aa(real,real,log(A2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),A2)) ).

% log_def
tff(fact_2751_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_2752_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_2753_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_2754_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2755_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = aa(num,num,bit0,inc(X)) ).

% inc.simps(3)
tff(fact_2756_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa(num,num,bit0,X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_2757_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_2758_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),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),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_2759_mult__inc,axiom,
    ! [X: num,Y4: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y4)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y4)),X) ).

% mult_inc
tff(fact_2760_log__base__change,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2761_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_2762_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: 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)),M))
     => ( 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,log(B2),M))) ) ) ).

% less_log_of_power
tff(fact_2763_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_2764_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_2765_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_2766_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_2767_log__mult,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != 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)),Y4))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y4)) ) ) ) ) ) ).

% log_mult
tff(fact_2768_log__divide,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != 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)),Y4))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y4)) ) ) ) ) ) ).

% log_divide
tff(fact_2769_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: 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)),M))
     => ( 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,log(B2),M))) ) ) ).

% le_log_of_power
tff(fact_2770_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),N)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_2771_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,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ).

% log_nat_power
tff(fact_2772_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_2773_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_2774_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_2775_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) )
     => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_2776_log__of__power__less,axiom,
    ! [M: 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),M)),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)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_2777_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != 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,log(A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2778_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_2779_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_2780_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_2781_arctan__ubound,axiom,
    ! [Y4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y4)),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_2782_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_2783_log__of__power__le,axiom,
    ! [M: 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),M)),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)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_2784_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_2785_arctan__lbound,axiom,
    ! [Y4: 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,Y4))) ).

% arctan_lbound
tff(fact_2786_arctan__bounded,axiom,
    ! [Y4: 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,Y4)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y4)),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_2787_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),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),M),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_2788_less__log2__of__power,axiom,
    ! [N: nat,M: 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)),M))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% less_log2_of_power
tff(fact_2789_le__log2__of__power,axiom,
    ! [N: nat,M: 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)),M))
     => 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,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% le_log2_of_power
tff(fact_2790_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),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),M),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_2791_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_2792_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_2793_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_2794_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_2795_sin__cos__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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_2796_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,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,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_2797_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2798_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,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2799_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,log(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,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),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_2800_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,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,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_2801_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_2802_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_2803_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_2804_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% floor_numeral
tff(fact_2805_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_2806_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% ceiling_numeral
tff(fact_2807_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_2808_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_2809_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_2810_sin__pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),X)) = sin(real,X) ).

% sin_pi_minus
tff(fact_2811_cos__minus__pi,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),pi)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_minus_pi
tff(fact_2812_cos__pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),X)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_pi_minus
tff(fact_2813_sin__minus__pi,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),pi)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_minus_pi
tff(fact_2814_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_2815_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,X))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_2816_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_2817_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_le_floor
tff(fact_2818_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_2819_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_2820_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_2821_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_numeral
tff(fact_2822_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_2823_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_2824_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_numeral
tff(fact_2825_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_2826_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_2827_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_2828_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_2829_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_2830_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_2831_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_2832_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_2833_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_2834_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_2835_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_2836_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_2837_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_2838_sin__npi2,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = zero_zero(real) ).

% sin_npi2
tff(fact_2839_sin__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_2840_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_2841_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,X))),A2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A2))) ) ).

% nat_ceiling_le_eq
tff(fact_2842_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_2843_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_2844_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_2845_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),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),V)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_2846_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> 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),V)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_2847_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_2848_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_2849_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> 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),V)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_2850_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),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),V)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_2851_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_2852_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V)))) ) ) ).

% floor_less_neg_numeral
tff(fact_2853_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2854_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2855_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_2856_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_2857_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2858_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_2859_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = cos(real,X) ).

% cos_periodic
tff(fact_2860_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = sin(real,X) ).

% sin_periodic
tff(fact_2861_cos__2pi__minus,axiom,
    ! [X: 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)) = cos(real,X) ).

% cos_2pi_minus
tff(fact_2862_cos__npi2,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi2
tff(fact_2863_cos__npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi
tff(fact_2864_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_2865_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_2866_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_2867_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_2868_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_2869_sin__2npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2870_cos__2npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2871_sin__2pi__minus,axiom,
    ! [X: 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),sin(real,X)) ).

% sin_2pi_minus
tff(fact_2872_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_2873_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_2874_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2875_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2876_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_2877_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_2878_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_2879_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y4))) ) ).

% sin_add
tff(fact_2880_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_2881_polar__Ex,axiom,
    ! [X: real,Y4: real] :
    ? [R3: real,A4: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),R3),cos(real,A4)) )
      & ( Y4 = aa(real,real,aa(real,fun(real,real),times_times(real),R3),sin(real,A4)) ) ) ).

% polar_Ex
tff(fact_2882_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y4))) ) ).

% sin_diff
tff(fact_2883_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_2884_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y4))) ) ).

% cos_add
tff(fact_2885_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y4))) ) ).

% cos_diff
tff(fact_2886_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_2887_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_2888_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_2889_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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))),sin(A,X))),cos(A,X)) ) ).

% sin_double
tff(fact_2890_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
      & ( sin(real,Y) = sin(real,X) )
      & ( cos(real,Y) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_2891_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y4))) ) ) ).

% floor_mono
tff(fact_2892_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y4)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_2893_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),sin(real,X)),X)) ) ).

% sin_x_le_x
tff(fact_2894_sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),one_one(real))) ).

% sin_le_one
tff(fact_2895_cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,X)),one_one(real))) ).

% cos_le_one
tff(fact_2896_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),sin(real,X))),aa(real,real,abs_abs(real),X))) ).

% abs_sin_x_le_abs_x
tff(fact_2897_cos__arctan__not__zero,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_2898_sin__cos__le1,axiom,
    ! [X: real,Y4: 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),sin(real,X)),sin(real,Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,X)),cos(real,Y4))))),one_one(real))) ).

% sin_cos_le1
tff(fact_2899_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% sin_squared_eq
tff(fact_2900_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_squared_eq
tff(fact_2901_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: 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,Y4))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)))) ) ).

% le_floor_add
tff(fact_2902_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2))))) ) ).

% of_nat_ceiling
tff(fact_2903_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)),sin(real,X))) ) ) ).

% sin_gt_zero
tff(fact_2904_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)),sin(real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_2905_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)),sin(real,X))) ) ) ).

% sin_ge_zero
tff(fact_2906_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_2907_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: 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),Y4))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y4)))) ) ).

% ceiling_add_le
tff(fact_2908_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))),sin(real,X))) ).

% sin_ge_minus_one
tff(fact_2909_cos__monotone__0__pi__le,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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),cos(real,X)),cos(real,Y4))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_2910_cos__mono__le__eq,axiom,
    ! [X: real,Y4: 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)),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,X)),cos(real,Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_2911_cos__inj__pi,axiom,
    ! [X: real,Y4: 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)),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),pi))
           => ( ( cos(real,X) = cos(real,Y4) )
             => ( X = Y4 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_2912_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))),cos(real,X))) ).

% cos_ge_minus_one
tff(fact_2913_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),sin(real,X))),one_one(real))) ).

% abs_sin_le_one
tff(fact_2914_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),cos(real,X))),one_one(real))) ).

% abs_cos_le_one
tff(fact_2915_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_sin
tff(fact_2916_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_cos
tff(fact_2917_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_sin
tff(fact_2918_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_plus_sin
tff(fact_2919_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_diff_sin
tff(fact_2920_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_diff_cos
tff(fact_2921_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),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),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_double
tff(fact_2922_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),W)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% cos_double_sin
tff(fact_2923_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R2))
         => 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,R2)))),R2)) ) ) ).

% of_nat_floor
tff(fact_2924_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: 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,A2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B2)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))))) ) ).

% le_mult_nat_floor
tff(fact_2925_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,N: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) ).

% floor_divide_of_nat_eq
tff(fact_2926_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_2927_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_2928_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_2929_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2)))) ) ).

% le_nat_floor
tff(fact_2930_cos__monotone__0__pi,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),cos(real,X)),cos(real,Y4))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_2931_cos__mono__less__eq,axiom,
    ! [X: real,Y4: 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)),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,X)),cos(real,Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_2932_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))
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_2933_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))
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_2934_cos__monotone__minus__pi__0_H,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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),cos(real,Y4)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_2935_sincos__total__pi,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ( ( 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),pi))
            & ( X = cos(real,T4) )
            & ( Y4 = sin(real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_2936_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,X)))
     => ( 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),cos(real,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_2937_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% sin_expansion_lemma
tff(fact_2938_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_2939_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% cos_expansion_lemma
tff(fact_2940_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)),sin(real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_2941_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_2942_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_2943_cos__two__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_less_zero
tff(fact_2944_cos__two__le__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_le_zero
tff(fact_2945_cos__is__zero,axiom,
    ? [X3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( cos(real,X3) = zero_zero(real) )
      & ! [Y3: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( cos(real,Y3) = zero_zero(real) ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
tff(fact_2946_cos__monotone__minus__pi__0,axiom,
    ! [Y4: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),cos(real,Y4)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_2947_cos__total,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),pi))
            & ( cos(real,X3) = Y4 )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),pi))
                  & ( cos(real,Y3) = Y4 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_2948_sincos__total__pi__half,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( ( 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
         => ? [T4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
              & ( X = cos(real,T4) )
              & ( Y4 = sin(real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_2949_sincos__total__2pi__le,axiom,
    ! [X: real,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
          & ( X = cos(real,T4) )
          & ( Y4 = sin(real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_2950_sincos__total__2pi,axiom,
    ! [X: real,Y4: 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ~ ! [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( ( X = cos(real,T4) )
               => ( Y4 != sin(real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_2951_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)),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_2952_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_45
tff(fact_2953_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_45
tff(fact_2954_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_cos
tff(fact_2955_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_plus_cos
tff(fact_2956_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)),sin(real,X))) ) ) ).

% sin_gt_zero2
tff(fact_2957_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),sin(real,X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_2958_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),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_2959_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_30
tff(fact_2960_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)),cos(real,X))) ) ) ).

% cos_gt_zero
tff(fact_2961_sin__monotone__2pi__le,axiom,
    ! [Y4: 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))))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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),sin(real,Y4)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_2962_sin__mono__le__eq,axiom,
    ! [X: real,Y4: 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))))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),sin(real,Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_2963_sin__inj__pi,axiom,
    ! [X: real,Y4: 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))))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( ( sin(real,X) = sin(real,Y4) )
             => ( X = Y4 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_2964_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_60
tff(fact_2965_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_60
tff(fact_2966_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_30
tff(fact_2967_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),W)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_2968_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),cos(A,X))) ) ).

% cos_treble_cos
tff(fact_2969_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),sin(real,X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_2970_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),sin(real,X)),zero_zero(real))) ) ) ).

% sin_less_zero
tff(fact_2971_sin__monotone__2pi,axiom,
    ! [Y4: 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))))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),sin(real,Y4)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_2972_sin__mono__less__eq,axiom,
    ! [X: real,Y4: 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))))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),sin(real,Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_2973_sin__total,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ? [X3: 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))))),X3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
            & ( sin(real,X3) = Y4 )
            & ! [Y3: 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))))),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
                  & ( sin(real,Y3) = Y4 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_2974_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)),cos(real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_2975_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)),cos(real,X))) ) ) ).

% cos_ge_zero
tff(fact_2976_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X5: 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),X5)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X5: 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),X5)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_2977_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)),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_2978_sin__arctan,axiom,
    ! [X: real] : sin(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% sin_arctan
tff(fact_2979_cos__arctan,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% cos_arctan
tff(fact_2980_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,log(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,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2981_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,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2982_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( 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_2983_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
            & ( 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),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2984_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( 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_2985_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
            & ( 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),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2986_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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_2987_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)))))
     => ( 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_2988_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)))))
     => ( 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_2989_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( Z != complex2(cos(real,T4),sin(real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_2990_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,log(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(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),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_2991_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)))
       => ( 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_2992_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_2993_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_2994_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_2995_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_2996_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_2997_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),powr(real,X,A2)))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_2998_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,A2,X)),zero_zero(real)))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_2999_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_3000_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( ( powr(real,A2,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3001_powr__one__gt__zero__iff,axiom,
    ! [X: 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_3002_powr__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_3003_powr__le__cancel__iff,axiom,
    ! [X: real,A2: 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),powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_3004_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),N)) ).

% numeral_powr_numeral_real
tff(fact_3005_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( powr(real,A2,aa(real,real,log(A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_3006_log__powr__cancel,axiom,
    ! [A2: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y4)) = Y4 ) ) ) ).

% log_powr_cancel
tff(fact_3007_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_3008_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_3009_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_3010_sin__arcsin,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ( sin(real,aa(real,real,arcsin,Y4)) = Y4 ) ) ) ).

% sin_arcsin
tff(fact_3011_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N)) ) ) ).

% powr_numeral
tff(fact_3012_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_3013_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_3014_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_3015_square__powr__half,axiom,
    ! [X: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),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_3016_powr__powr,axiom,
    ! [X: real,A2: real,B2: real] : powr(real,powr(real,X,A2),B2) = powr(real,X,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_3017_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = zero_zero(complex) )
    <=> ( ( A2 = zero_zero(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_3018_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_3019_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),D2)) ).

% complex_diff
tff(fact_3020_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),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),Y4))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,Y4,A2)),powr(real,X,A2))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3021_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,A2,X)),zero_zero(real))) ).

% powr_non_neg
tff(fact_3022_powr__ge__pzero,axiom,
    ! [X: real,Y4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),powr(real,X,Y4))) ).

% powr_ge_pzero
tff(fact_3023_powr__mono2,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( 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),Y4))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,Y4,A2))) ) ) ) ).

% powr_mono2
tff(fact_3024_powr__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_mono
tff(fact_3025_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = one_one(complex) )
    <=> ( ( A2 = one_one(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_3026_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3027_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_3028_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( 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),Y4))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,Y4,A2))) ) ) ) ).

% powr_less_mono2
tff(fact_3029_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),Y4))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,Y4,A2)),powr(real,X,A2))) ) ) ) ).

% powr_mono2'
tff(fact_3030_gr__one__powr,axiom,
    ! [X: real,Y4: 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)),Y4))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),powr(real,X,Y4))) ) ) ).

% gr_one_powr
tff(fact_3031_powr__inj,axiom,
    ! [A2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,X) = powr(real,A2,Y4) )
        <=> ( X = Y4 ) ) ) ) ).

% powr_inj
tff(fact_3032_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: 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)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),powr(real,X,A2))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3033_powr__mono__both,axiom,
    ! [A2: real,B2: real,X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),Y4))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,Y4,B2))) ) ) ) ) ).

% powr_mono_both
tff(fact_3034_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( 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),powr(real,X,A2)),one_one(real))) ) ) ) ).

% powr_le1
tff(fact_3035_powr__divide,axiom,
    ! [X: real,Y4: real,A2: 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)),Y4))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A2)),powr(real,Y4,A2)) ) ) ) ).

% powr_divide
tff(fact_3036_powr__mult,axiom,
    ! [X: real,Y4: real,A2: 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)),Y4))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A2)),powr(real,Y4,A2)) ) ) ) ).

% powr_mult
tff(fact_3037_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),powr(real,B2,C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),powr(real,B2,aa(real,real,uminus_uminus(real),C2))) ).

% divide_powr_uminus
tff(fact_3038_ln__powr,axiom,
    ! [X: real,Y4: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,X,Y4)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y4),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_powr
tff(fact_3039_log__base__powr,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( A2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A2,B2)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),B2) ) ) ).

% log_base_powr
tff(fact_3040_log__powr,axiom,
    ! [X: real,B2: real,Y4: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,X,Y4)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y4),aa(real,real,log(B2),X)) ) ) ).

% log_powr
tff(fact_3041_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_3042_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_3043_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A,B2: A] : powr(A,X,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,X,A2)),powr(A,X,B2)) ) ).

% powr_add
tff(fact_3044_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z1),Z22)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),powr(A,W,Z1)),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_3045_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_3046_arcsin__le__arcsin,axiom,
    ! [X: real,Y4: 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),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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,Y4))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3047_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_3048_arcsin__le__mono,axiom,
    ! [X: real,Y4: 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),Y4)),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,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ).

% arcsin_le_mono
tff(fact_3049_arcsin__eq__iff,axiom,
    ! [X: real,Y4: 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),Y4)),one_one(real)))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y4) )
        <=> ( X = Y4 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3050_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,tan(A),X4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,X4)),cos(A,X4)) ) ).

% tan_def
tff(fact_3051_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N) ) ) ).

% powr_realpow
tff(fact_3052_powr__less__iff,axiom,
    ! [B2: real,X: real,Y4: 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),powr(real,B2,Y4)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),aa(real,real,log(B2),X))) ) ) ) ).

% powr_less_iff
tff(fact_3053_less__powr__iff,axiom,
    ! [B2: real,X: real,Y4: 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),powr(real,B2,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),X)),Y4)) ) ) ) ).

% less_powr_iff
tff(fact_3054_log__less__iff,axiom,
    ! [B2: real,X: real,Y4: 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,log(B2),X)),Y4))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,Y4))) ) ) ) ).

% log_less_iff
tff(fact_3055_less__log__iff,axiom,
    ! [B2: real,X: real,Y4: 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),Y4),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,Y4)),X)) ) ) ) ).

% less_log_iff
tff(fact_3056_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A] : powr(A,X,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,X,A2)) ) ).

% powr_minus_divide
tff(fact_3057_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X) ) ) ).

% powr_neg_one
tff(fact_3058_arcsin__less__arcsin,axiom,
    ! [X: real,Y4: 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),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y4))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3059_powr__mult__base,axiom,
    ! [X: real,Y4: 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),powr(real,X,Y4)) = powr(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y4)) ) ) ).

% powr_mult_base
tff(fact_3060_arcsin__less__mono,axiom,
    ! [X: real,Y4: 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),Y4)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ).

% arcsin_less_mono
tff(fact_3061_le__log__iff,axiom,
    ! [B2: real,X: real,Y4: 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),Y4),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,Y4)),X)) ) ) ) ).

% le_log_iff
tff(fact_3062_log__le__iff,axiom,
    ! [B2: real,X: real,Y4: 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,log(B2),X)),Y4))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,Y4))) ) ) ) ).

% log_le_iff
tff(fact_3063_le__powr__iff,axiom,
    ! [B2: real,X: real,Y4: 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),powr(real,B2,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),X)),Y4)) ) ) ) ).

% le_powr_iff
tff(fact_3064_powr__le__iff,axiom,
    ! [B2: real,X: real,Y4: 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),powr(real,B2,Y4)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,log(B2),X))) ) ) ) ).

% powr_le_iff
tff(fact_3065_ln__powr__bound,axiom,
    ! [X: real,A2: 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)),A2))
       => 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),powr(real,X,A2)),A2))) ) ) ).

% ln_powr_bound
tff(fact_3066_ln__powr__bound2,axiom,
    ! [X: real,A2: 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)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),X),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),X))) ) ) ).

% ln_powr_bound2
tff(fact_3067_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_3068_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y4: 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,log(B2),X)),Y4) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y4))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3069_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y4: 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),Y4),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y4)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3070_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y4: 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),Y4),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y4)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3071_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_3072_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)))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3073_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A2: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,A2) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,A2) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),X))) ) ) ) ) ).

% powr_def
tff(fact_3074_lemma__tan__total,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),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),Y4),aa(real,real,tan(real),X3))) ) ) ).

% lemma_tan_total
tff(fact_3075_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_3076_lemma__tan__total1,axiom,
    ! [Y4: real] :
    ? [X3: 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))))),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),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),X3) = Y4 ) ) ).

% lemma_tan_total1
tff(fact_3077_tan__mono__lt__eq,axiom,
    ! [X: real,Y4: 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))))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3078_tan__monotone_H,axiom,
    ! [Y4: 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))))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),Y4),X))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y4)),aa(real,real,tan(real),X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3079_tan__monotone,axiom,
    ! [Y4: 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))))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),Y4)),aa(real,real,tan(real),X))) ) ) ) ).

% tan_monotone
tff(fact_3080_tan__total,axiom,
    ! [Y4: real] :
    ? [X3: 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))))),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),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),X3) = Y4 )
      & ! [Y3: 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))))),Y3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),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),Y3) = Y4 ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
tff(fact_3081_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_3082_tan__inverse,axiom,
    ! [Y4: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y4)) = 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)))),Y4)) ).

% tan_inverse
tff(fact_3083_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y4: 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,log(B2),X)),Y4) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,aa(real,real,uminus_uminus(real),Y4)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3084_complex__norm,axiom,
    ! [X: real,Y4: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y4)) = 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_norm
tff(fact_3085_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y4) != 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),Y4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y4))) ) ) ) ) ).

% add_tan_eq
tff(fact_3086_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_3087_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_3088_tan__total__pos,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),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),X3) = Y4 ) ) ) ).

% tan_total_pos
tff(fact_3089_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_3090_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_3091_tan__mono__le,axiom,
    ! [X: real,Y4: 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),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),Y4))) ) ) ) ).

% tan_mono_le
tff(fact_3092_tan__mono__le__eq,axiom,
    ! [X: real,Y4: 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))))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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),Y4)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3093_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_3094_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_3095_arctan__unique,axiom,
    ! [X: real,Y4: 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) = Y4 )
         => ( aa(real,real,arctan,Y4) = X ) ) ) ) ).

% arctan_unique
tff(fact_3096_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_3097_arctan,axiom,
    ! [Y4: 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,Y4)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y4)),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,Y4)) = Y4 ) ) ).

% arctan
tff(fact_3098_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y4) != 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),Y4))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y4))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3099_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y4) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = 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),Y4))),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),Y4)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3100_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y4) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = 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),Y4))),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),Y4)))) ) ) ) ) ) ).

% tan_add
tff(fact_3101_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)))
     => ? [Z2: 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)))))),Z2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),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),Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_3102_arcsin__lt__bounded,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),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,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,Y4)),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_3103_arcsin__lbound,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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,Y4))) ) ) ).

% arcsin_lbound
tff(fact_3104_arcsin__ubound,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y4)),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_3105_arcsin__bounded,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y4)),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_3106_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,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3107_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3108_arcsin,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & ( sin(real,aa(real,real,arcsin,Y4)) = Y4 ) ) ) ) ).

% arcsin
tff(fact_3109_arcsin__pi,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),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,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y4)),pi))
          & ( sin(real,aa(real,real,arcsin,Y4)) = Y4 ) ) ) ) ).

% arcsin_pi
tff(fact_3110_arcsin__le__iff,axiom,
    ! [X: real,Y4: 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)))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),Y4))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),sin(real,Y4))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3111_le__arcsin__iff,axiom,
    ! [X: real,Y4: 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)))),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),aa(real,real,arcsin,X)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,Y4)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3112_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,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_3113_sin__arccos__abs,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y4)),one_one(real)))
     => ( sin(real,aa(real,real,arccos,Y4)) = 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),Y4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3114_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)))
       => ( 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_3115_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),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_3116_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))
       => ( 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))
       => ( 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_3117_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_3118_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N4: int] : X = aa(int,A,ring_1_of_int(A),N4) ) ) ).

% of_int_floor_cancel
tff(fact_3119_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N4: int] : X = aa(int,A,ring_1_of_int(A),N4) ) ) ).

% of_int_ceiling_cancel
tff(fact_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y4: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4)) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y4)) ) ).

% of_real_mult
tff(fact_3135_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : real_Vector_of_real(A,aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_3136_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real,Y4: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y4)) ) ).

% of_real_divide
tff(fact_3137_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_3138_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_3139_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y4: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y4)) ) ).

% of_real_diff
tff(fact_3140_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_3141_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_3142_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_3143_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_3144_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_3145_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3146_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_3147_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,real_Vector_of_real(A,pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_3148_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_3149_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_3150_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,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y4: 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),Y4) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y4 ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_3167_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y4: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y4) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y4 = 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_3168_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_3169_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_3170_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_3171_sin__npi__int,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3172_cos__arccos,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ( cos(real,aa(real,real,arccos,Y4)) = Y4 ) ) ) ).

% cos_arccos
tff(fact_3173_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_3174_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),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),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_3175_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: 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),A2)))
        <=> 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)),A2)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_3176_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),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),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_3177_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: 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),A2)))
        <=> 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)),A2)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_3178_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y4: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y4) = 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) )
        <=> ( Y4 = 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_3179_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y4: 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),Y4) )
        <=> ( 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) = Y4 ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_3180_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,X)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_3181_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_3182_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3183_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3184_sin__int__2pin,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3185_cos__int__2pin,axiom,
    ! [N: int] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ).

% cos_int_2pin
tff(fact_3186_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: 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),A2)))
        <=> 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)),A2)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_3187_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_3188_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: 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),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_3189_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: 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),A2)))
        <=> 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)),A2)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_3190_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ).

% ex_le_of_int
tff(fact_3191_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_3192_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% cos_int_times_real
tff(fact_3193_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% sin_int_times_real
tff(fact_3194_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_3195_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_3196_complex__of__real__def,axiom,
    ! [R2: real] : real_Vector_of_real(complex,R2) = complex2(R2,zero_zero(real)) ).

% complex_of_real_def
tff(fact_3197_complex__of__real__code,axiom,
    ! [X4: real] : real_Vector_of_real(complex,X4) = complex2(X4,zero_zero(real)) ).

% complex_of_real_code
tff(fact_3198_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y4: real,Xa: real] :
      ( ( complex2(X,Y4) = real_Vector_of_real(complex,Xa) )
    <=> ( ( X = Xa )
        & ( Y4 = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_3199_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y4: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),complex2(X,Y4)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),X),aa(real,real,aa(real,fun(real,real),times_times(real),R2),Y4)) ).

% complex_of_real_mult_Complex
tff(fact_3200_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y4: real,R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(X,Y4)),real_Vector_of_real(complex,R2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),X),R2),aa(real,real,aa(real,fun(real,real),times_times(real),Y4),R2)) ).

% Complex_mult_complex_of_real
tff(fact_3201_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_of_int
tff(fact_3202_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_3203_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_int_mask_eq
tff(fact_3204_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y4: real,X: real] :
          ( ( Y4 != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y4)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3205_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_3206_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_3207_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A2)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),A2)) ) ) ).

% ceiling_le
tff(fact_3208_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_3209_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_3210_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_3211_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_3212_arccos__le__arccos,axiom,
    ! [X: real,Y4: 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),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y4)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_3213_arccos__eq__iff,axiom,
    ! [X: real,Y4: 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),Y4)),one_one(real))) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y4) )
      <=> ( X = Y4 ) ) ) ).

% arccos_eq_iff
tff(fact_3214_arccos__le__mono,axiom,
    ! [X: real,Y4: 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),Y4)),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,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),X)) ) ) ) ).

% arccos_le_mono
tff(fact_3215_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_3216_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_3217_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_3218_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_3219_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),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),X3),one_one(int)))))
          & ! [Y3: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),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),Y3),one_one(int))))) )
             => ( Y3 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_3220_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),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),Z2),one_one(int))))) ) ) ).

% floor_exists
tff(fact_3221_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: 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,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A)))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3222_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_3223_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: 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,R2))),one_one(A))),R2)) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3224_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_3225_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),M))
    <=> 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),M)),one_one(real)))) ) ).

% int_le_real_less
tff(fact_3226_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),M))
    <=> 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),M))) ) ).

% int_less_real_le
tff(fact_3227_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_3228_sin__zero__iff__int2,axiom,
    ! [X: 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_3229_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Theta),aa(real,real,aa(real,fun(real,real),times_times(real),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_3230_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_3231_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_3232_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real)))) ).

% real_of_int_floor_add_one_gt
tff(fact_3233_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real)))) ).

% real_of_int_floor_add_one_ge
tff(fact_3234_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3235_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: 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),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3236_arccos__lbound,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y4))) ) ) ).

% arccos_lbound
tff(fact_3237_arccos__less__arccos,axiom,
    ! [X: real,Y4: 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),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y4)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_3238_arccos__less__mono,axiom,
    ! [X: real,Y4: 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),Y4)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),X)) ) ) ) ).

% arccos_less_mono
tff(fact_3239_arccos__ubound,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y4)),pi)) ) ) ).

% arccos_ubound
tff(fact_3240_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,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3241_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_3242_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,X) = A2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),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),A2)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_3243_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_3244_cos__arccos__abs,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y4)),one_one(real)))
     => ( cos(real,aa(real,real,arccos,Y4)) = Y4 ) ) ).

% cos_arccos_abs
tff(fact_3245_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: 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,A2)))),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)))) ) ).

% norm_of_real_diff
tff(fact_3246_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_3247_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archimedean_ceiling(A,X) = A2 )
        <=> ( 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),A2)),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),A2))) ) ) ) ).

% ceiling_eq_iff
tff(fact_3248_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_3249_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_3250_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,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3251_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_3252_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_3253_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_3254_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_3255_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_3256_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_3257_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_3258_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_3259_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: 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),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3260_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => 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),P3),Q2)))),Q2)),P3)) ) ) ).

% floor_divide_lower
tff(fact_3261_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),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),P3),Q2)))),Q2))) ) ) ).

% ceiling_divide_upper
tff(fact_3262_arccos__lt__bounded,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y4)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_3263_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(int,A,ring_1_of_int(A),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_of_int_iff
tff(fact_3264_arccos__bounded,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y4)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_3265_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)))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3266_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,cos(real,X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3267_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_3268_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_3269_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P3),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),P3),Q2)))),one_one(A))),Q2))) ) ) ).

% floor_divide_upper
tff(fact_3270_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => 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),P3),Q2)))),one_one(A))),Q2)),P3)) ) ) ).

% ceiling_divide_lower
tff(fact_3271_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_3272_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X5: 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),X5)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3273_arccos,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y4)),pi))
          & ( cos(real,aa(real,real,arccos,Y4)) = Y4 ) ) ) ) ).

% arccos
tff(fact_3274_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_3275_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,X) = 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_3276_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,X) = 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_3277_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),sin(A,X)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3278_arccos__le__pi2,axiom,
    ! [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y4)),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_3279_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,log(B2),X)) = K )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(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),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_3280_cos__zero__iff__int,axiom,
    ! [X: 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_3281_sin__zero__iff__int,axiom,
    ! [X: 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_3282_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))
         => ( 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))
         => ( 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_3283_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: 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),Y4)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),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) = Y4 ) ) ) ) ).

% round_unique
tff(fact_3284_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_3285_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_3286_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_3287_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_3288_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_3289_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_3290_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_3291_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_3292_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y4))) ) ) ).

% round_mono
tff(fact_3293_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_3294_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_3295_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: 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),M))))) ) ).

% round_diff_minimal
tff(fact_3296_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_3297_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_3298_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_3299_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_3300_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_3301_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_3302_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_3303_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_3304_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_3305_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_3306_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_3307_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_3308_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_3309_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_3310_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_3311_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3312_Complex__eq__i,axiom,
    ! [X: real,Y4: real] :
      ( ( complex2(X,Y4) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y4 = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3313_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_3314_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,cot(A),X4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,X4)),sin(A,X4)) ) ).

% cot_def
tff(fact_3315_complex__of__real__i,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),imaginary_unit) = complex2(zero_zero(real),R2) ).

% complex_of_real_i
tff(fact_3316_i__complex__of__real,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,R2)) = complex2(zero_zero(real),R2) ).

% i_complex_of_real
tff(fact_3317_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_3318_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y4: 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,Y4))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y4)) ) )
          & ( ~ 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,Y4))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = 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,Y4))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),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_3326_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% csqrt_eq_0
tff(fact_3327_csqrt__0,axiom,
    csqrt(zero_zero(complex)) = zero_zero(complex) ).

% csqrt_0
tff(fact_3328_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3329_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_3330_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_3331_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)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3332_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_3333_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(2)
tff(fact_3334_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_3335_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(3)
tff(fact_3336_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_3337_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),csqrt(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z ).

% power2_csqrt
tff(fact_3338_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% bit_0
tff(fact_3339_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_3340_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_3341_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),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))),A2)) ) ) ) ).

% bit_mod_2_iff
tff(fact_3342_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_3343_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_3344_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( cis(arg(Z)) = aa(complex,complex,sgn_sgn(complex),Z) ) ) ).

% cis_Arg
tff(fact_3345_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),M)),N)) ) ) ).

% bit_numeral_iff
tff(fact_3346_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N)) ) ) ).

% bit_of_nat_iff_bit
tff(fact_3347_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_3348_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_and_iff
tff(fact_3349_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_and_int_iff
tff(fact_3350_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & ( M != N ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_3351_cis__divide,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cis(A2)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)) ).

% cis_divide
tff(fact_3352_cis__neq__zero,axiom,
    ! [A2: real] : cis(A2) != zero_zero(complex) ).

% cis_neq_zero
tff(fact_3353_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_3354_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_3355_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_3356_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_3357_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_3358_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_3359_DeMoivre,axiom,
    ! [A2: real,N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),N) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) ).

% DeMoivre
tff(fact_3360_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_3361_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_3362_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),N),A2) = aa(A,A,aa(nat,fun(A,A),if(fun(nat,fun(A,A)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),N),A2) ) ).

% flip_bit_eq_if
tff(fact_3363_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_3364_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_3365_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( 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)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3366_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_3367_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,bit_concat_bit(N,K),aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_3368_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3369_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)) ) ) ).

% bit_Suc
tff(fact_3370_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_3371_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_3372_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M2))
              <=> 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_3373_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(aa(A,bool,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),A2),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_3374_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3375_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_3376_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( real_Vector_of_real(complex,aa(real,real,sqrt,X)) = csqrt(real_Vector_of_real(complex,X)) ) ) ).

% of_real_sqrt
tff(fact_3377_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3378_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3379_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_3380_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [J2: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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))),A2)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3381_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_3382_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% set_bit_eq
tff(fact_3383_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,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_3384_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_3385_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_3386_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))
         => ( 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))
         => ( 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_3387_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_3388_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_3389_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3390_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ).

% inverse_inverse_eq
tff(fact_3391_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_3392_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y4)) = Y4 ) ).

% bit.xor_left_self
tff(fact_3393_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_3394_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3395_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_3396_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3397_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_3398_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ).

% inverse_divide
tff(fact_3399_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ).

% inverse_minus_eq
tff(fact_3400_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_3401_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_3402_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_3403_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_3404_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ).

% abs_inverse
tff(fact_3405_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ).

% inverse_sgn
tff(fact_3406_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A2)) ) ).

% sgn_inverse
tff(fact_3407_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_xor
tff(fact_3408_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3409_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3410_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% inverse_positive_iff_positive
tff(fact_3411_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_3412_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3413_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3414_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_3415_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3416_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_3417_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_3418_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3419_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( 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),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3420_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_3421_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_3422_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_3423_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_3424_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_3425_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,bit0,Y4))) = 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),Y4))) ) ).

% xor_numerals(3)
tff(fact_3426_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y4)) ) ).

% xor_numerals(1)
tff(fact_3427_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y4)) ) ).

% xor_numerals(2)
tff(fact_3428_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_3429_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_3430_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y4))) = 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),Y4))) ) ).

% xor_numerals(7)
tff(fact_3431_xor__nat__numerals_I1_J,axiom,
    ! [Y4: 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,Y4))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y4)) ).

% xor_nat_numerals(1)
tff(fact_3432_xor__nat__numerals_I2_J,axiom,
    ! [Y4: 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,Y4))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y4)) ).

% xor_nat_numerals(2)
tff(fact_3433_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_3434_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_3435_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% xor_numerals(4)
tff(fact_3436_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% xor_numerals(6)
tff(fact_3437_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),ring_1_Ints(A))) ) ) ).

% Ints_power
tff(fact_3438_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real] :
          ( ( X != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,inverse_inverse(real),X)) = aa(A,A,inverse_inverse(A),real_Vector_of_real(A,X)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_3439_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_3440_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_3441_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_3442_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_mult
tff(fact_3443_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_3444_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_3445_real__sqrt__inverse,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ).

% real_sqrt_inverse
tff(fact_3446_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_diff
tff(fact_3447_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,uminus_uminus(A),A2)),ring_1_Ints(A))) ) ) ).

% Ints_minus
tff(fact_3448_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_3449_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_3450_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_3451_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_3452_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_3453_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( A2 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_3454_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_xor_eq
tff(fact_3455_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_3456_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_3457_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_3458_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_3459_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_3460_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y4: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y4),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y4)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y4)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3461_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_3462_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_3463_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y4: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y4),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y4),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_xor_distrib2
tff(fact_3464_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y4),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y4)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_3465_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),N))
        <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_xor_iff
tff(fact_3466_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,abs_abs(A),A2)),ring_1_Ints(A))) ) ) ).

% Ints_abs
tff(fact_3467_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_3468_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Q2),ring_1_Ints(A)))
         => ( ! [Z2: int] : pp(aa(A,bool,P,aa(int,A,ring_1_of_int(A),Z2)))
           => pp(aa(A,bool,P,Q2)) ) ) ) ).

% Ints_induct
tff(fact_3469_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Q2),ring_1_Ints(A)))
         => ~ ! [Z2: int] : Q2 != aa(int,A,ring_1_of_int(A),Z2) ) ) ).

% Ints_cases
tff(fact_3470_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R2))) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3471_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ).

% positive_imp_inverse_positive
tff(fact_3472_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_3473_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3474_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3475_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),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),A2))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3476_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),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),A2)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3477_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => 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),A2))) ) ) ) ).

% less_imp_inverse_less
tff(fact_3478_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% inverse_less_imp_less
tff(fact_3479_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3480_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_3481_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_3482_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3483_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_3484_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_3485_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_3486_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_3487_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_3488_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ).

% inverse_eq_divide
tff(fact_3489_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_3490_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3491_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_3492_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3493_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_3494_divide__real__def,axiom,
    ! [X: real,Y4: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y4)) ).

% divide_real_def
tff(fact_3495_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3496_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_3497_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3498_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),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),A2)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3499_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => 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),A2))) ) ) ) ).

% le_imp_inverse_le
tff(fact_3500_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% inverse_le_imp_le
tff(fact_3501_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_3502_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_3503_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_less_inverse
tff(fact_3504_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3505_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3506_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_3507_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3508_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_3509_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3510_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3511_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_3512_inverse__powr,axiom,
    ! [Y4: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4))
     => ( powr(real,aa(real,real,inverse_inverse(real),Y4),A2) = aa(real,real,inverse_inverse(real),powr(real,Y4,A2)) ) ) ).

% inverse_powr
tff(fact_3513_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_3514_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A2))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))),ring_1_Ints(A))) ) ) ).

% of_int_divide_in_Ints
tff(fact_3515_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),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),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_3516_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),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),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_3517_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_3518_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_3519_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2))) ) ) ) ).

% one_le_inverse
tff(fact_3520_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3521_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_3522_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3523_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3524_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2))
         => 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),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3525_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3526_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3527_forall__pos__mono__1,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D3: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E))
         => ( pp(aa(real,bool,P,D3))
           => 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_3528_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
    <=> ? [N4: nat] :
          ( ( N4 != 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),N4))))
          & 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),N4))),E2)) ) ) ).

% real_arch_inverse
tff(fact_3529_forall__pos__mono,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D3: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E))
         => ( pp(aa(real,bool,P,D3))
           => 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_3530_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_3531_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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),bit_se5824344971392196577ns_xor(A),A2),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_xor_iff
tff(fact_3532_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_3533_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_3534_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(A,bool,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)),A2)),A2)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_3535_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_3536_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_3537_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y4: 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),Y4),ring_1_Ints(A)))
           => ( ( X = Y4 )
            <=> 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),Y4))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3538_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),pi)) = zero_zero(real) )
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),ring_1_Ints(real))) ) ).

% sin_times_pi_eq_0
tff(fact_3539_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_3540_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3541_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_3542_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3543_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3544_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_3545_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_3546_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),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),M),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_3547_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_3548_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3549_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_3550_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3551_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_3552_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,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,A2)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_3553_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: A] :
          ( ( archimedean_frac(A,X) = A2 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_3554_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,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),A2),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3555_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3556_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_3557_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( 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,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3558_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3559_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3560_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3561_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_3562_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_3563_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_3564_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),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),M),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_3565_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_3566_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,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))),M))),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),M),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_3567_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(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))),A2)))),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))),A2)))) ) ).

% one_xor_eq
tff(fact_3568_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(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))),A2)))),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))),A2)))) ) ).

% xor_one_eq
tff(fact_3569_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_3570_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ) ).

% tan_sec
tff(fact_3571_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,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_3572_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_3573_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_3574_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_3575_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_3576_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_3577_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_3578_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_3579_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3580_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_3581_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2) ) ).

% push_bit_push_bit
tff(fact_3582_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_and
tff(fact_3583_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_xor
tff(fact_3584_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa(int,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_3585_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( sinh(real,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_3586_sinh__real__le__iff,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sinh(real,X)),sinh(real,Y4)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ).

% sinh_real_le_iff
tff(fact_3587_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cosh(A,zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_3588_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(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_3589_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(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_3590_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(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_3591_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(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_3592_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_3593_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_3594_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_3595_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_3596_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_numeral
tff(fact_3597_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_3598_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_3599_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_3600_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: 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),A2)))
        <=> ( ( 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))),A2)) ) ) ) ).

% even_push_bit_iff
tff(fact_3601_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_3602_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,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_3603_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),N))
    <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_3604_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y4))) ) ).

% sinh_add
tff(fact_3605_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y4))) ) ).

% cosh_add
tff(fact_3606_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : cosh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y4))) ) ).

% cosh_diff
tff(fact_3607_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sinh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y4))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y4))) ) ).

% sinh_diff
tff(fact_3608_cosh__real__nonzero,axiom,
    ! [X: real] : cosh(real,X) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_3609_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) ).

% push_bit_nat_eq
tff(fact_3610_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sinh(real,X)),cosh(real,X))) ).

% sinh_le_cosh_real
tff(fact_3611_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M)) ) ).

% push_bit_of_nat
tff(fact_3612_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),N)) = aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_push_bit
tff(fact_3613_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) ) ).

% push_bit_minus
tff(fact_3614_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_add
tff(fact_3615_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) ) ).

% push_bit_of_int
tff(fact_3616_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sinh(A,X)),cosh(A,X)) ) ).

% tanh_def
tff(fact_3617_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),sinh(A,X)),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_3618_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),cosh(A,X)),sinh(A,X)) = aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)) ) ).

% cosh_minus_sinh
tff(fact_3619_XOR__lower,axiom,
    ! [X: int,Y4: 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)),Y4))
       => 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),Y4))) ) ) ).

% XOR_lower
tff(fact_3620_cosh__real__pos,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cosh(real,X))) ).

% cosh_real_pos
tff(fact_3621_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y4: 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),Y4),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cosh(real,X)),cosh(real,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_3622_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cosh(real,X)),cosh(real,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_3623_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),cosh(real,X))) ).

% cosh_real_nonneg
tff(fact_3624_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),cosh(real,X))) ).

% cosh_real_ge_1
tff(fact_3625_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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))),sinh(A,X))),cosh(A,X)) ) ).

% sinh_double
tff(fact_3626_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(A,A,bit_se4730199178511100633sh_bit(A,M),A2)) ) ).

% push_bit_take_bit
tff(fact_3627_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),A2)) ) ).

% take_bit_push_bit
tff(fact_3628_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_3629_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y4: 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),Y4),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3630_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y4: 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)),Y4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y4)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3631_cosh__real__strict__mono,axiom,
    ! [X: real,Y4: 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),Y4))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y4))) ) ) ).

% cosh_real_strict_mono
tff(fact_3632_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,M),K)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3633_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_3634_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_3635_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),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),sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_3636_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),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),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_3637_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),Q2)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3638_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),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_3639_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),L)) ).

% concat_bit_eq
tff(fact_3640_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_3641_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),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),sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cosh_double
tff(fact_3642_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_3643_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3644_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se4730199178511100633sh_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_int_def
tff(fact_3645_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% push_bit_nat_def
tff(fact_3646_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_3647_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: 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)),A2))
         => ~ ! [B3: A] : A2 != aa(A,A,bit_se4730199178511100633sh_bit(A,N),B3) ) ) ).

% exp_dvdE
tff(fact_3648_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_3649_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y4) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = 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),Y4))),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),Y4)))) ) ) ) ) ).

% tanh_add
tff(fact_3650_XOR__upper,axiom,
    ! [X: int,N: nat,Y4: 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),Y4),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),Y4)),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_3651_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cosh_field_def
tff(fact_3652_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% complex_inverse
tff(fact_3653_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_3654_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sinh_field_def
tff(fact_3655_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_3656_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_3657_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
     => vEBT_invar_vebt(T2,D2) ) ).

% valid_eq2
tff(fact_3658_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(T2,D2)
     => vEBT_VEBT_valid(T2,D2) ) ).

% valid_eq1
tff(fact_3659_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
    <=> vEBT_invar_vebt(T2,D2) ) ).

% valid_eq
tff(fact_3660_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_3661_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),N: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( pp(aa(nat,bool,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_3662_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y4) )
        <=> ( X = Y4 ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_3663_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = X ) ).

% bit.double_compl
tff(fact_3664_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y4)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y4)) ) ).

% bit.xor_compl_right
tff(fact_3665_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y4) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y4)) ) ).

% bit.xor_compl_left
tff(fact_3666_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_3667_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_3668_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_3669_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_3670_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_3671_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_3672_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_3673_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_3674_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_3675_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_3676_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_3677_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: 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_ri4277139882892585799ns_not(A),A2)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% even_not_iff
tff(fact_3678_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_3679_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_3680_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_3681_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_3682_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_3683_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff
tff(fact_3684_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_3685_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_3686_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_diff_distrib
tff(fact_3687_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_3688_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_3689_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_3690_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_3691_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_3692_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2)) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_3693_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,N)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3694_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_3695_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int)))) ).

% unset_bit_int_def
tff(fact_3696_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_3697_even__not__iff__int,axiom,
    ! [K: 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,bit_ri4277139882892585799ns_not(int),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_not_iff_int
tff(fact_3698_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_3699_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(4)
tff(fact_3700_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_3701_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_3702_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3703_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_3704_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_3705_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),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),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),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(5)
tff(fact_3706_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(7)
tff(fact_3707_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_3708_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(6)
tff(fact_3709_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(9)
tff(fact_3710_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_3711_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_3712_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),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),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_3713_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_3714_Cauchy__iff2,axiom,
    ! [X7: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X7)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
         => ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N4))
             => 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,X7,M5)),aa(nat,real,X7,N4)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_3715_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_3716_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_3717_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sinh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),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_3718_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
        <=> ( ( A2 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3719_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : aa(A,A,real_V8093663219630862766scaleR(A,A2),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_3720_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [X: A,A2: real,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4)) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)) ) ).

% mult_scaleR_right
tff(fact_3721_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),Y4) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4)) ) ).

% mult_scaleR_left
tff(fact_3722_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y4: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4) )
        <=> ( ( X = Y4 )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_3723_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),X) ) ).

% scaleR_scaleR
tff(fact_3724_complex__Re__of__nat,axiom,
    ! [N: nat] : re(aa(nat,complex,semiring_1_of_nat(complex),N)) = aa(nat,real,semiring_1_of_nat(real),N) ).

% complex_Re_of_nat
tff(fact_3725_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_3726_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3727_complex__Re__numeral,axiom,
    ! [V: num] : re(aa(num,complex,numeral_numeral(complex),V)) = aa(num,real,numeral_numeral(real),V) ).

% complex_Re_numeral
tff(fact_3728_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_3729_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,X: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A2)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_scaleR
tff(fact_3730_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_3731_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,R2))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),R2) ).

% Re_divide_of_real
tff(fact_3732_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_3733_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A2) ) ).

% scaleR_times
tff(fact_3734_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_3735_cos__Arg__i__mult__zero,axiom,
    ! [Y4: complex] :
      ( ( Y4 != zero_zero(complex) )
     => ( ( re(Y4) = zero_zero(real) )
       => ( cos(real,arg(Y4)) = zero_zero(real) ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_3736_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% inverse_scaleR_times
tff(fact_3737_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% fraction_scaleR_times
tff(fact_3738_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_3739_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A2: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3740_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y4: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4)) ) ).

% scaleR_right_diff_distrib
tff(fact_3741_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] : re(aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(X)) ).

% scaleR_complex.simps(1)
tff(fact_3742_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y4: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4) )
           => ( X = Y4 ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_3743_real__scaleR__def,axiom,
    ! [A2: real,X: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),X) ).

% real_scaleR_def
tff(fact_3744_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,R2),X) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R2)),X) ) ).

% scaleR_conv_of_real
tff(fact_3745_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ).

% scaleR_left_diff_distrib
tff(fact_3746_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Y4: real,Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4)),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,X),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,Y4),Xa)) ) ).

% scaleR_left.diff
tff(fact_3747_imaginary__unit_Osimps_I1_J,axiom,
    re(imaginary_unit) = zero_zero(real) ).

% imaginary_unit.simps(1)
tff(fact_3748_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(X)),real_V7770717601297561774m_norm(complex,X))) ).

% complex_Re_le_cmod
tff(fact_3749_zero__complex_Osimps_I1_J,axiom,
    re(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(1)
tff(fact_3750_complex__scaleR,axiom,
    ! [R2: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),B2)) ).

% complex_scaleR
tff(fact_3751_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y4: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y4)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),re(X)),re(Y4)) ).

% minus_complex.simps(1)
tff(fact_3752_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X))) ) ) ) ).

% scaleR_right_mono
tff(fact_3753_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),A2))
         => ( 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,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3754_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3755_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3756_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
        <=> ( ( pp(aa(real,bool,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),A2),B2)) )
            & ( 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),B2),A2)) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3757_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y4: A,A2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4))) ) ) ) ).

% scaleR_left_mono
tff(fact_3758_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),zero_zero(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3759_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) = X )
        <=> ( ( ( V = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),X) ) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_3760_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V: real,A2: A] :
          ( ( X = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) )
        <=> ( ( ( V = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,V),X) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2) ) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_3761_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E2)),C2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3762_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),E2)),D2))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3763_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),real_V7770717601297561774m_norm(complex,X))) ).

% abs_Re_le_cmod
tff(fact_3764_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(csqrt(Z)))) ).

% Re_csqrt
tff(fact_3765_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & 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),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3766_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & 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),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3767_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A,Y4: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => ( 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),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y4))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3768_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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)),A2))
             => ( 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,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3769_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & 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),A2),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),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_3770_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & 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),A2),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)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_3771_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( 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)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3772_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( 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),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3773_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3774_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3775_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A2: 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),A2),one_one(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3776_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% scaleR_2
tff(fact_3777_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y4: A,X: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y4)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3778_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,X: A,C2: A,Y4: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) = Y4 )
          <=> ( X = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y4)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3779_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3780_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,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,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3781_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3782_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,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),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3783_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3784_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3785_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,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,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3786_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3787_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,X: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3788_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3789_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3790_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3791_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3792_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3793_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3794_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: 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),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3795_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: 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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3796_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_3797_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A2: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) = re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_3798_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ? [M8: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M2))
               => ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N7))
                   => 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,X7,M2)),aa(nat,A,X7,N7)))),E2)) ) ) ) ) ) ).

% CauchyD
tff(fact_3799_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M4))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),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,X7,M4)),aa(nat,A,X7,N2)))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI
tff(fact_3800_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [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))
                 => ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),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,X7,M5)),aa(nat,A,X7,N4)))),E3)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3801_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cosh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),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_3802_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_3803_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_3804_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(X)),zero_zero(real)))
       => ( csqrt(X) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sqrt,aa(real,real,abs_abs(real),re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_3805_Complex__divide,axiom,
    ! [X: complex,Y4: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y4) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y4)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y4)),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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y4)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% Complex_divide
tff(fact_3806_complex__Im__of__int,axiom,
    ! [Z: int] : im(aa(int,complex,ring_1_of_int(complex),Z)) = zero_zero(real) ).

% complex_Im_of_int
tff(fact_3807_complex__Im__fact,axiom,
    ! [N: nat] : im(semiring_char_0_fact(complex,N)) = zero_zero(real) ).

% complex_Im_fact
tff(fact_3808_Im__complex__of__real,axiom,
    ! [Z: real] : im(real_Vector_of_real(complex,Z)) = zero_zero(real) ).

% Im_complex_of_real
tff(fact_3809_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),X),N)) = zero_zero(real) ) ) ).

% Im_power_real
tff(fact_3810_complex__Im__numeral,axiom,
    ! [V: num] : im(aa(num,complex,numeral_numeral(complex),V)) = zero_zero(real) ).

% complex_Im_numeral
tff(fact_3811_complex__Im__of__nat,axiom,
    ! [N: nat] : im(aa(nat,complex,semiring_1_of_nat(complex),N)) = zero_zero(real) ).

% complex_Im_of_nat
tff(fact_3812_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,R2))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),R2) ).

% Im_divide_of_real
tff(fact_3813_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_3814_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),X),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),re(X)),N) ) ) ).

% Re_power_real
tff(fact_3815_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_3816_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_3817_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(X)))
       => ( csqrt(X) = real_Vector_of_real(complex,aa(real,real,sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_3818_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_3819_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] : im(aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(X)) ).

% scaleR_complex.simps(2)
tff(fact_3820_zero__complex_Osimps_I2_J,axiom,
    im(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(2)
tff(fact_3821_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_3822_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y4: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y4)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),im(X)),im(Y4)) ).

% minus_complex.simps(2)
tff(fact_3823_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Z),ring_1_Ints(complex)))
    <=> ( ( im(Z) = zero_zero(real) )
        & ? [I2: int] : re(Z) = aa(int,real,ring_1_of_int(real),I2) ) ) ).

% complex_is_Int_iff
tff(fact_3824_scaleR__complex_Ocode,axiom,
    ! [R2: real,X: complex] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),X) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(X)),aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(X))) ).

% scaleR_complex.code
tff(fact_3825_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),real_V7770717601297561774m_norm(complex,X))) ).

% abs_Im_le_cmod
tff(fact_3826_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y4: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y4)) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y4))) ).

% times_complex.simps(2)
tff(fact_3827_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( im(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = aa(real,real,abs_abs(real),re(Z)) ) ) ).

% cmod_eq_Re
tff(fact_3828_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( re(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = aa(real,real,abs_abs(real),im(Z)) ) ) ).

% cmod_eq_Im
tff(fact_3829_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa(real,real,abs_abs(real),re(Z)) = real_V7770717601297561774m_norm(complex,Z) )
     => ( im(Z) = zero_zero(real) ) ) ).

% Im_eq_0
tff(fact_3830_cmod__Im__le__iff,axiom,
    ! [X: complex,Y4: complex] :
      ( ( re(X) = re(Y4) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),aa(real,real,abs_abs(real),im(Y4)))) ) ) ).

% cmod_Im_le_iff
tff(fact_3831_cmod__Re__le__iff,axiom,
    ! [X: complex,Y4: complex] :
      ( ( im(X) = im(Y4) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),aa(real,real,abs_abs(real),re(Y4)))) ) ) ).

% cmod_Re_le_iff
tff(fact_3832_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y4: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y4)) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y4))) ).

% times_complex.simps(1)
tff(fact_3833_minus__complex_Ocode,axiom,
    ! [X: complex,Y4: complex] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y4) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),re(X)),re(Y4)),aa(real,real,aa(real,fun(real,real),minus_minus(real),im(X)),im(Y4))) ).

% minus_complex.code
tff(fact_3834_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(csqrt(Z))))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(csqrt(Z)))) ) ) ).

% csqrt_principal
tff(fact_3835_cmod__le,axiom,
    ! [Z: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z))))) ).

% cmod_le
tff(fact_3836_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A2: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) = im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_3837_Re__exp,axiom,
    ! [Z: complex] : re(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),cos(real,im(Z))) ).

% Re_exp
tff(fact_3838_Im__exp,axiom,
    ! [Z: complex] : im(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),sin(real,im(Z))) ).

% Im_exp
tff(fact_3839_times__complex_Ocode,axiom,
    ! [X: complex,Y4: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y4) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y4))),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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y4)))) ).

% times_complex.code
tff(fact_3840_cmod__power2,axiom,
    ! [Z: complex] : aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% cmod_power2
tff(fact_3841_Im__power2,axiom,
    ! [X: complex] : im(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),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(X))),im(X)) ).

% Im_power2
tff(fact_3842_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_3843_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_3844_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% norm_complex_def
tff(fact_3845_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_3846_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> 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,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% complex_neq_0
tff(fact_3847_Re__divide,axiom,
    ! [X: complex,Y4: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y4)) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y4)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Re_divide
tff(fact_3848_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),W),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(W)))
          | ( ( re(W) = zero_zero(real) )
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(W))) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_3849_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(B2)))
        | ( ( re(B2) = zero_zero(real) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(B2))) ) )
     => ( csqrt(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = B2 ) ) ).

% csqrt_square
tff(fact_3850_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_3851_Im__divide,axiom,
    ! [X: complex,Y4: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y4)) = 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(Y4))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y4)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y4)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Im_divide
tff(fact_3852_complex__abs__le__norm,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),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),real_V7770717601297561774m_norm(complex,Z)))) ).

% complex_abs_le_norm
tff(fact_3853_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_3854_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_3855_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R2),real_Vector_Reals(complex)))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R2),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(R2))),im(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Im_Reals_divide
tff(fact_3856_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X3) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_3857_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R2),real_Vector_Reals(complex)))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R2),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R2)),re(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Re_Reals_divide
tff(fact_3858_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_3859_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% complex_cnj_zero_iff
tff(fact_3860_complex__cnj__zero,axiom,
    cnj(zero_zero(complex)) = zero_zero(complex) ).

% complex_cnj_zero
tff(fact_3861_complex__cnj__diff,axiom,
    ! [X: complex,Y4: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y4)) = aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),cnj(X)),cnj(Y4)) ).

% complex_cnj_diff
tff(fact_3862_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = zero_zero(real) ).

% complex_In_mult_cnj_zero
tff(fact_3863_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R2),real_Vector_Reals(complex)))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),R2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),re(R2)) ) ) ).

% Re_divide_Reals
tff(fact_3864_imaginary__eq__real__iff,axiom,
    ! [Y4: complex,X: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Y4),real_Vector_Reals(complex)))
     => ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),X),real_Vector_Reals(complex)))
       => ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y4) = X )
        <=> ( ( X = zero_zero(complex) )
            & ( Y4 = zero_zero(complex) ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_3865_real__eq__imaginary__iff,axiom,
    ! [Y4: complex,X: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Y4),real_Vector_Reals(complex)))
     => ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),X),real_Vector_Reals(complex)))
       => ( ( X = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y4) )
        <=> ( ( X = zero_zero(complex) )
            & ( Y4 = zero_zero(complex) ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_3866_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R2),real_Vector_Reals(complex)))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),R2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),re(R2)) ) ) ).

% Im_divide_Reals
tff(fact_3867_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_3868_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_divide
tff(fact_3869_Reals__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_diff
tff(fact_3870_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),times_times(A),A2),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_mult
tff(fact_3871_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),real_Vector_Reals(A))) ) ).

% Reals_numeral
tff(fact_3872_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_3873_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Z),real_Vector_Reals(complex)))
    <=> ( im(Z) = zero_zero(real) ) ) ).

% complex_is_Real_iff
tff(fact_3874_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),real_Vector_Reals(A))) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_3875_Complex__in__Reals,axiom,
    ! [X: real] : pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),complex2(X,zero_zero(real))),real_Vector_Reals(complex))) ).

% Complex_in_Reals
tff(fact_3876_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_Reals(A)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,inverse_inverse(A),A2)),real_Vector_Reals(A))) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_3877_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_3878_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_3879_Re__complex__div__gt__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2))))) ) ).

% Re_complex_div_gt_0
tff(fact_3880_Re__complex__div__lt__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_lt_0
tff(fact_3881_Re__complex__div__ge__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2))))) ) ).

% Re_complex_div_ge_0
tff(fact_3882_Re__complex__div__le__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_le_0
tff(fact_3883_Im__complex__div__gt__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2))))) ) ).

% Im_complex_div_gt_0
tff(fact_3884_Im__complex__div__lt__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_lt_0
tff(fact_3885_Im__complex__div__ge__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2))))) ) ).

% Im_complex_div_ge_0
tff(fact_3886_Im__complex__div__le__0,axiom,
    ! [A2: 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),A2),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),A2),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_le_0
tff(fact_3887_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% complex_mod_mult_cnj
tff(fact_3888_complex__div__gt__0,axiom,
    ! [A2: 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),A2),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),A2),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),A2),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),A2),cnj(B2))))) ) ) ).

% complex_div_gt_0
tff(fact_3889_complex__norm__square,axiom,
    ! [Z: complex] : real_Vector_of_real(complex,aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) ).

% complex_norm_square
tff(fact_3890_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(Z))) ).

% complex_add_cnj
tff(fact_3891_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))),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_3892_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z)),W)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_3893_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_mult_cnj
tff(fact_3894_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_3895_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_3896_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% or_numerals(7)
tff(fact_3897_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% or_numerals(6)
tff(fact_3898_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_3899_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_3900_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_3901_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_3902_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_3903_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_3904_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2) ) ).

% drop_bit_drop_bit
tff(fact_3905_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_or
tff(fact_3906_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_or
tff(fact_3907_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_and
tff(fact_3908_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_or
tff(fact_3909_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_xor
tff(fact_3910_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_3911_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_3912_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_3913_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_3914_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_3915_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y4)) ) ).

% bit.de_Morgan_disj
tff(fact_3916_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y4)) ) ).

% bit.de_Morgan_conj
tff(fact_3917_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_3918_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_3919_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_3920_or__nat__numerals_I2_J,axiom,
    ! [Y4: 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,Y4))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y4)) ).

% or_nat_numerals(2)
tff(fact_3921_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_3922_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y4)) ) ).

% or_numerals(2)
tff(fact_3923_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_3924_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_3925_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_3926_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_3927_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_3928_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_3929_or__nat__numerals_I1_J,axiom,
    ! [Y4: 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,Y4))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y4)) ).

% or_nat_numerals(1)
tff(fact_3930_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_3931_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,bit0,Y4))) = 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),Y4))) ) ).

% or_numerals(3)
tff(fact_3932_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y4: 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,Y4))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y4)) ) ).

% or_numerals(1)
tff(fact_3933_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_3934_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_3935_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_3936_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_3937_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_3938_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_3939_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_3940_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_3941_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),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_3942_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),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_3943_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y4: 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,Y4))) = 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),Y4)))) ) ).

% or_numerals(4)
tff(fact_3944_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_3945_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_3946_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_3947_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_3948_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y4),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y4)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_3949_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y4),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y4)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_3950_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y4: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y4),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y4),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_disj_distrib2
tff(fact_3951_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y4: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y4),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y4),X)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),X)) ) ).

% bit.disj_conj_distrib2
tff(fact_3952_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_3953_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_3954_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_or_int_iff
tff(fact_3955_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_or_iff
tff(fact_3956_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),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),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_3957_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,M),N)) = aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_drop_bit
tff(fact_3958_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M)) ) ).

% drop_bit_of_nat
tff(fact_3959_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_or_eq
tff(fact_3960_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_3961_OR__lower,axiom,
    ! [X: int,Y4: 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)),Y4))
       => 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),Y4))) ) ) ).

% OR_lower
tff(fact_3962_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_3963_plus__and__or,axiom,
    ! [X: int,Y4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y4)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y4)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y4) ).

% plus_and_or
tff(fact_3964_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_3965_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_3966_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_3967_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_3968_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = A2 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_3969_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,M),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),M),N)),aa(int,int,bit_se4730199178511100633sh_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_3970_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2)) ) ).

% take_bit_drop_bit
tff(fact_3971_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),aa(A,A,bit_se4197421643247451524op_bit(A,M),A2)) ) ).

% drop_bit_take_bit
tff(fact_3972_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_3973_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y4) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y4)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y4))) ) ).

% bit.xor_def2
tff(fact_3974_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y4) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y4))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y4)) ) ).

% bit.xor_def
tff(fact_3975_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_3976_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_3977_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_3978_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),L)) ).

% concat_bit_def
tff(fact_3979_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_3980_set__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int))) ).

% set_bit_int_def
tff(fact_3981_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = A2 ) ).

% bits_ident
tff(fact_3982_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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),bit_se1065995026697491101ons_or(A),A2),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_or_iff
tff(fact_3983_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y4) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y4) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y4 ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3984_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_3985_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_3986_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_3987_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_3988_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y4) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y4) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y4 ) ) ) ) ).

% bit.compl_unique
tff(fact_3989_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_3990_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_3991_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_3992_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_3993_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% drop_bit_Suc
tff(fact_3994_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% drop_bit_eq_div
tff(fact_3995_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: 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_se4197421643247451524op_bit(A,N),A2)))
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_3996_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2))) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_3997_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_3998_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% or_not_numerals(6)
tff(fact_3999_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% or_one_eq
tff(fact_4000_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% one_or_eq
tff(fact_4001_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_4002_OR__upper,axiom,
    ! [X: int,N: nat,Y4: 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),Y4),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),Y4)),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_4003_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),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),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_4004_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_4005_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_4006_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_4007_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_4008_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = A2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = 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),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_4009_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_4010_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),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),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_4011_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),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),M),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_4012_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_4013_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_4014_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_4015_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_4016_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.bounded_iff
tff(fact_4017_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_4018_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_4019_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_4020_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_4021_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_4022_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_4023_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_4024_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_4025_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_4026_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_4027_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_4028_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_4029_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_4030_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_4031_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y4: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y4)) = 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),Y4)) ) ).

% of_nat_max
tff(fact_4032_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ).

% max_diff_distrib_left
tff(fact_4033_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y4: 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),Y4),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_4034_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ).

% max_add_distrib_left
tff(fact_4035_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4) = Y4 ) ) ) ).

% max_absorb2
tff(fact_4036_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4) = X ) ) ) ).

% max_absorb1
tff(fact_4037_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ) ).

% max_def
tff(fact_4038_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: 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),A2),B2))) ) ) ).

% max.coboundedI2
tff(fact_4039_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => 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),A2),B2))) ) ) ).

% max.coboundedI1
tff(fact_4040_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_4041_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_4042_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y4: 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),Y4)))
        <=> ( 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),Y4)) ) ) ) ).

% le_max_iff_disj
tff(fact_4043_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: 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),A2),B2))) ) ).

% max.cobounded2
tff(fact_4044_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ).

% max.cobounded1
tff(fact_4045_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_4046_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => 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)),A2)) ) ) ) ).

% max.boundedI
tff(fact_4047_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.boundedE
tff(fact_4048_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% max.orderI
tff(fact_4049_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_4050_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( 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),A2),B2))) ) ) ) ).

% max.mono
tff(fact_4051_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_4052_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y4: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y4)) = 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),Y4)) ) ).

% of_int_max
tff(fact_4053_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit0,N),one2) = aa(num,num,bit0,one2) ).

% or_not_num_neg.simps(4)
tff(fact_4054_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4197421643247451524op_bit(int,N),K)) ).

% drop_bit_nat_eq
tff(fact_4055_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit0,N),aa(num,num,bit1,M)) = aa(num,num,bit0,bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(6)
tff(fact_4056_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit1,N),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_4057_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_4058_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit0,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_4059_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% drop_bit_nat_def
tff(fact_4060_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_4061_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_4062_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_4063_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_4064_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_4065_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M,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),M),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_4066_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_4067_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_4068_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_4069_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).

% max_Suc_Suc
tff(fact_4070_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_4071_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_4072_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_4073_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4074_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_4075_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4076_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_4077_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_4078_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_4079_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_4080_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_4081_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_max_right
tff(fact_4082_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_4083_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ).

% nat_add_max_right
tff(fact_4084_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_4085_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_4086_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).

% nat_minus_add_max
tff(fact_4087_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit0,N)) = aa(num,num,bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_4088_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,N)) ).

% semiring_norm(28)
tff(fact_4089_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit0,N),aa(num,num,bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(5)
tff(fact_4090_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa(num,num,bit0,N) ).

% inc_BitM_eq
tff(fact_4091_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(9)
tff(fact_4092_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_4093_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_4094_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_4095_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_4096_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(8)
tff(fact_4097_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_4098_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),bitM(W)))) ) ).

% odd_numeral_BitM
tff(fact_4099_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_4100_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y4: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y4 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y4 != one2 ) ) )
       => ( ( ( X = one2 )
           => ! [M4: num] :
                ( ( Xa = aa(num,num,bit0,M4) )
               => ( Y4 != aa(num,num,bit1,M4) ) ) )
         => ( ( ( X = one2 )
             => ! [M4: num] :
                  ( ( Xa = aa(num,num,bit1,M4) )
                 => ( Y4 != aa(num,num,bit1,M4) ) ) )
           => ( ( ? [N2: num] : X = aa(num,num,bit0,N2)
               => ( ( Xa = one2 )
                 => ( Y4 != aa(num,num,bit0,one2) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa(num,num,bit0,N2) )
                   => ! [M4: num] :
                        ( ( Xa = aa(num,num,bit0,M4) )
                       => ( Y4 != bitM(bit_or_not_num_neg(N2,M4)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa(num,num,bit0,N2) )
                     => ! [M4: num] :
                          ( ( Xa = aa(num,num,bit1,M4) )
                         => ( Y4 != aa(num,num,bit0,bit_or_not_num_neg(N2,M4)) ) ) )
                 => ( ( ? [N2: num] : X = aa(num,num,bit1,N2)
                     => ( ( Xa = one2 )
                       => ( Y4 != one2 ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa(num,num,bit1,N2) )
                         => ! [M4: num] :
                              ( ( Xa = aa(num,num,bit0,M4) )
                             => ( Y4 != bitM(bit_or_not_num_neg(N2,M4)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa(num,num,bit1,N2) )
                           => ! [M4: num] :
                                ( ( Xa = aa(num,num,bit1,M4) )
                               => ( Y4 != bitM(bit_or_not_num_neg(N2,M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_4101_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_4102_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) = 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_4103_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_4104_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_4105_real__root__zero,axiom,
    ! [N: nat] : aa(real,real,root(N),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4106_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_4107_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_4108_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y4: 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),Y4) )
      <=> ( X = Y4 ) ) ) ).

% real_root_eq_iff
tff(fact_4109_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_4110_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,M,N) ) ).

% diff_numeral_simps(1)
tff(fact_4111_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_4112_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_4113_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V)),Y4) ) ).

% semiring_norm(167)
tff(fact_4114_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W)),Y4) ) ).

% semiring_norm(166)
tff(fact_4115_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_4116_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4117_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_4118_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4)) ) ) ).

% real_root_less_iff
tff(fact_4119_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M) ) ).

% diff_numeral_simps(4)
tff(fact_4120_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y4)) ) ) ).

% real_root_le_iff
tff(fact_4121_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_4122_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_4123_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_4124_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_4125_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_4126_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).

% diff_numeral_special(2)
tff(fact_4127_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_4128_real__root__gt__0__iff,axiom,
    ! [N: nat,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4)) ) ) ).

% real_root_gt_0_iff
tff(fact_4129_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_4130_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_4131_real__root__ge__0__iff,axiom,
    ! [N: nat,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y4)) ) ) ).

% real_root_ge_0_iff
tff(fact_4132_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_4133_real__root__gt__1__iff,axiom,
    ! [N: nat,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y4)) ) ) ).

% real_root_gt_1_iff
tff(fact_4134_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_4135_real__root__ge__1__iff,axiom,
    ! [N: nat,Y4: 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),Y4)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y4)) ) ) ).

% real_root_ge_1_iff
tff(fact_4136_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_4137_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_4138_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_4139_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_4140_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_4141_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(2)
tff(fact_4142_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(1)
tff(fact_4143_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).

% minus_sub_one_diff_one
tff(fact_4144_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_4145_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).

% diff_numeral_special(8)
tff(fact_4146_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_4147_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_4148_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_4149_real__root__inverse,axiom,
    ! [N: nat,X: real] : aa(real,real,root(N),aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,root(N),X)) ).

% real_root_inverse
tff(fact_4150_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),X) = aa(real,real,root(M),aa(real,real,root(N),X)) ).

% real_root_mult_exp
tff(fact_4151_real__root__mult,axiom,
    ! [N: nat,X: real,Y4: real] : aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y4)) ).

% real_root_mult
tff(fact_4152_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_4153_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] : aa(real,real,root(M),aa(real,real,root(N),X)) = aa(real,real,root(N),aa(real,real,root(M),X)) ).

% real_root_commute
tff(fact_4154_real__root__divide,axiom,
    ! [N: nat,X: real,Y4: real] : aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y4)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y4)) ).

% real_root_divide
tff(fact_4155_real__root__pos__pos__le,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),zero_zero(real)),aa(real,real,root(N),X))) ) ).

% real_root_pos_pos_le
tff(fact_4156_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_4157_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y4: 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),Y4))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y4))) ) ) ).

% real_root_less_mono
tff(fact_4158_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y4: 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),Y4))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y4))) ) ) ).

% real_root_le_mono
tff(fact_4159_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_4160_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_4161_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_4162_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% sub_non_positive
tff(fact_4163_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% sub_non_negative
tff(fact_4164_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% sub_negative
tff(fact_4165_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% sub_positive
tff(fact_4166_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_4167_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_4168_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: 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),N5))
       => ( 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(N5),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4169_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% sqrt_def
tff(fact_4170_root__abs__power,axiom,
    ! [N: nat,Y4: 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),Y4),N))) = aa(real,real,abs_abs(real),Y4) ) ) ).

% root_abs_power
tff(fact_4171_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_4172_real__root__strict__increasing,axiom,
    ! [N: nat,N5: 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),N5))
       => ( 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(N5),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4173_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_4174_real__root__decreasing,axiom,
    ! [N: nat,N5: 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),N5))
       => ( 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(N5),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4175_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_4176_real__root__pos__unique,axiom,
    ! [N: nat,Y4: 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)),Y4))
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),N) = X )
         => ( aa(real,real,root(N),X) = Y4 ) ) ) ) ).

% real_root_pos_unique
tff(fact_4177_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_4178_odd__real__root__pow,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))
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_4179_odd__real__root__unique,axiom,
    ! [N: nat,Y4: real,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))
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),N) = X )
       => ( aa(real,real,root(N),X) = Y4 ) ) ) ).

% odd_real_root_unique
tff(fact_4180_odd__real__root__power__cancel,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))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_4181_real__root__increasing,axiom,
    ! [N: nat,N5: 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),N5))
       => ( 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(N5),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4182_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_4183_root__sgn__power,axiom,
    ! [N: nat,Y4: 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),Y4)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y4)),N))) = Y4 ) ) ).

% root_sgn_power
tff(fact_4184_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_4185_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_4186_log__root,axiom,
    ! [N: nat,A2: 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)),A2))
       => ( aa(real,real,log(B2),aa(real,real,root(N),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A2)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_4187_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,log(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_4188_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))
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),N)) = X )
             => pp(aa(real,bool,P,Y5)) ) ) ) ) ).

% split_root
tff(fact_4189_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( ? [K8: real] :
            ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)))
         => real_V4916620083959148203axioms(A,B,F2) ) ) ).

% bounded_linear_axioms.intro
tff(fact_4190_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F2)
        <=> ? [K6: real] :
            ! [X5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K6))) ) ) ).

% bounded_linear_axioms_def
tff(fact_4191_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_4192_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_4193_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_4194_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_4195_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_4196_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_4197_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_4198_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_4199_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_4200_VEBT__internal_Ospace_H_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B3: bool] : X != vEBT_Leaf(A4,B3)
     => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X != vEBT_Node(Info2,Deg2,TreeList2,Summary2) ) ).

% VEBT_internal.space'.cases
tff(fact_4201_VEBT_Oexhaust,axiom,
    ! [Y4: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y4 != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X223: bool] : Y4 != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_4202_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_4203_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_4204_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_4205_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_4206_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_4207_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Leaf(A2,B2)) = one_one(real) ).

% VEBT_internal.cnt.simps(1)
tff(fact_4208_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_4209_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_4210_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A2,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_4211_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf(A2,B2)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_4212_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf(A2,B2)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_4213_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y4: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y4 )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y4 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_4214_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y4: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y4 )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y4 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_4215_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y4: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y4 )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y4 != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y4 != aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_4216_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_4217_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_4218_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y4: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y4 )
     => ( pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),X))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Leaf(A4,B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) )
                 => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2))) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_4219_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_4220_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_4221_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y4: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y4 )
     => ( pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel),X))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) )
               => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Leaf(A4,B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) )
                 => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2))) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_4222_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y4: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y4 )
     => ( pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),X))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y4 = one_one(real) )
               => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Leaf(A4,B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y4 = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) )
                 => ~ pp(aa(vEBT_VEBT,bool,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_4223_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_4224_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),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_4225_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,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(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),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 = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4226_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,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(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_4227_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),some(A,X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_4228_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M4: nat] : X != aa(nat,product_prod(nat,nat),product_Pair(nat,nat,K2),M4) ).

% prod_decode_aux.cases
tff(fact_4229_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),zero_zero(nat),Va2,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_4230_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_4231_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [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))),M)))
                   => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_12))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),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 )
                         => ! [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))),M)))
                             => ( ( ( 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))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,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(X3,N))) )
                                   => ( 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),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4232_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [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))),M)))
                   => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_12))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),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 )
                         => ! [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))),M)))
                             => ( ( ( 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))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,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(X3,N))) )
                                   => ( 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),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4233_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: bool,B3: bool] : A1 = vEBT_Leaf(A4,B3)
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_invar_vebt(X4,N2) )
                 => ( vEBT_invar_vebt(Summary2,M4)
                   => ( ( 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))),M4) )
                     => ( ( M4 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                         => ( ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),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),TreeList2)))
                                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_invar_vebt(X4,N2) )
                   => ( vEBT_invar_vebt(Summary2,M4)
                     => ( ( 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))),M4) )
                       => ( ( M4 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                           => ( ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),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),TreeList2)))
                                   => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_invar_vebt(X4,N2) )
                     => ( vEBT_invar_vebt(Summary2,M4)
                       => ( ( 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))),M4) )
                         => ( ( M4 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                             => ( ! [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))),M4)))
                                   => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                       => ~ ? [X_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),Mi2),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)))
                                     => ~ ( ( Mi2 != 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))),M4)))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                & ! [X4: nat] :
                                                    ( ( ( vEBT_VEBT_high(X4,N2) = I4 )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N2))) )
                                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X4))
                                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                           => vEBT_invar_vebt(X4,N2) )
                       => ( vEBT_invar_vebt(Summary2,M4)
                         => ( ( 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))),M4) )
                           => ( ( M4 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                               => ( ! [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))),M4)))
                                     => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                         => ~ ? [X_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),Mi2),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)))
                                       => ~ ( ( Mi2 != 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))),M4)))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                  & ! [X4: nat] :
                                                      ( ( ( vEBT_VEBT_high(X4,N2) = I4 )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N2))) )
                                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X4))
                                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4234_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A6: bool,B7: bool] : A1 = vEBT_Leaf(A6,B7)
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X5,N4) )
            & vEBT_invar_vebt(Summary3,N4)
            & ( 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))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_12)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X5,N4) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N4))
            & ( 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,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_12)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X5,N4) )
            & vEBT_invar_vebt(Summary3,N4)
            & ( 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))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ! [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))),N4)))
               => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_12))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_12)) ) )
            & 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))),N4)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N4) = 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,N4))) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N4) = 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(X5,N4))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X5))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X5,N4) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N4))
            & ( 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,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ! [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,N4))))
               => ( ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_12))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_12)) ) )
            & 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,N4))))
                 => ( ( ( vEBT_VEBT_high(Ma3,N4) = 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,N4))) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N4) = 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(X5,N4))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X5))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4235_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R2: A,Q2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2)) = 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))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),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)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2)) = 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))),Q2)),R2) ) ) ) ) ).

% divmod_step_eq
tff(fact_4236_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_4237_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: 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)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),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)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_4238_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X2: A] : size_option(A,X,some(A,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_4239_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_4240_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,D3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),D3)
     => ~ ! [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_4241_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),product_Pair(num,num,one2),one2) )
     => ( ! [N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2))
       => ( ! [N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2))
         => ( ! [M4: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),one2)
           => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit0,N2))
             => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit1,N2))
               => ( ! [M4: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),one2)
                 => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit0,N2))
                   => ~ ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4242_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B3: bool,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A4,B3)),X3)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_4243_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),Uy2: 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,Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X3)
         => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),X3)
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT,X3: 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)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_4244_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_option(A,X,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_4245_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_4246_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),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,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_4247_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: 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),A2),one_one(int)),B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4248_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_4249_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% numeral_div_minus_numeral
tff(fact_4250_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_div_numeral
tff(fact_4251_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,N,M)) ) ) ).

% dvd_numeral_simp
tff(fact_4252_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),product_Pair(A,A,aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_4253_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_4254_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_4255_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_4256_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_4257_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),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),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_4258_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_4259_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_4260_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_4261_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_4262_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( modulo_modulo(int,K,L) = R2 ) ) ).

% mod_int_unique
tff(fact_4263_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4264_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_4265_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,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),M)),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_4266_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,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),M)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_4267_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,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),M)),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_4268_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,aa(int,product_prod(int,int),product_Pair(int,int,if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),aa(int,int,uminus_uminus(int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_4269_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & ( pp(aa(int,bool,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)),R2))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),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),R2))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q2 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4270_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = 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),R2)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4271_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,Q3: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q3: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),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),Q3),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4272_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) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),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),Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4273_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_4274_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4275_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),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
                 => ( ( Xa = Mi2 )
                    | ( 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_4276_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y4) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y4) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => pp(Y4) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( pp(Y4)
                <=> ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
                 => ( pp(Y4)
                  <=> ~ ( ( Xa = Mi2 )
                        | ( 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(Y4)
                    <=> ~ ( ( 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_4277_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_4278_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y4: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y4 )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y4 = 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))
         => ( Y4 = 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_4279_foldr__same,axiom,
    ! [Xs: list(real),Y4: real] :
      ( ! [X3: real,Y: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),aa(list(real),set(real),set2(real),Xs)))
         => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y),aa(list(real),set(real),set2(real),Xs)))
           => ( X3 = Y ) ) )
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),aa(list(real),set(real),set2(real),Xs)))
           => ( X3 = Y4 ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y4) ) ) ) ).

% foldr_same
tff(fact_4280_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D2: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(nat),nat,size_size(list(nat)),Xs)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,nth(nat,Xs),I3)),aa(nat,nat,nth(nat,Ys),I3))) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),D2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),foldr(nat,nat,plus_plus(nat),Xs,C2)),aa(list(nat),nat,size_size(list(nat)),Ys))),foldr(nat,nat,plus_plus(nat),Ys,D2))) ) ) ) ).

% foldr_mono
tff(fact_4281_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X3)),Bound)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4282_foldr__zero,axiom,
    ! [Xs: list(nat),D2: nat] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(nat),nat,size_size(list(nat)),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),foldr(nat,nat,plus_plus(nat),Xs,D2)),D2))) ) ).

% foldr_zero
tff(fact_4283_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D2: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X3)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X3)))) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),foldr(real,real,plus_plus(real),map(A,real,G,Xs),zero_zero(real)))),D2))) ) ).

% f_g_map_foldr_bound
tff(fact_4284_real__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),C2)) = foldr(real,real,plus_plus(real),map(A,real,aTP_Lamp_aa(fun(A,nat),fun(A,real),F2),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_4285_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X3)),Bound)) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4286_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Xa2) = Xa2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Xa2) = X4 ) ) ) ) ).

% max_def_raw
tff(fact_4287_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_4288_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ) ).

% less_set_def
tff(fact_4289_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))),A3)) ).

% Collect_subset
tff(fact_4290_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ) ).

% less_eq_set_def
tff(fact_4291_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: 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),A2))),collect(A,aTP_Lamp_ac(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4292_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: 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),A2))),collect(A,aTP_Lamp_ac(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_4293_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_ad(set(A),fun(A,bool),A3)) ).

% Compl_eq
tff(fact_4294_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : collect(A,aTP_Lamp_ae(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_4295_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3))) ).

% uminus_set_def
tff(fact_4296_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_af(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_4297_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ag(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_4298_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ah(set(A),fun(set(A),fun(A,bool)),A3),B4)) ).

% set_diff_eq
tff(fact_4299_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = 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),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ).

% minus_set_def
tff(fact_4300_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_ai(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_4301_nat__leq__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_leq_as_int
tff(fact_4302_nat__less__as__int,axiom,
    ! [X4: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_less_as_int
tff(fact_4303_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_4304_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_even
tff(fact_4305_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_odd
tff(fact_4306_nat__plus__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_plus_as_int
tff(fact_4307_nat__times__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_times_as_int
tff(fact_4308_nat__minus__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X4),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_minus_as_int
tff(fact_4309_nat__div__as__int,axiom,
    ! [X4: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_div_as_int
tff(fact_4310_nat__mod__as__int,axiom,
    ! [X4: nat,Xa2: nat] : modulo_modulo(nat,X4,Xa2) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X4),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_mod_as_int
tff(fact_4311_diff__nat__eq__if,axiom,
    ! [Z4: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = aa(int,nat,nat2,Z) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = 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),Z4)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4))) ) ) ) ).

% diff_nat_eq_if
tff(fact_4312_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_aj(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_4313_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)) ) ).

% signed_take_bit_code
tff(fact_4314_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ak(A,fun(nat,fun(A,A)),A2),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_4315_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( ( ( K = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_al(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K)) ) ) ) ) ).

% gbinomial_code
tff(fact_4316_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_4317_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [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_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_4318_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [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_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(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_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(nat)))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_4319_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,S2),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,V)),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,V)),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,V)),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,V)),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_4320_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y4: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y4 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) ) )
         => ~ ! [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))))
                     => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),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))))
                     => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_4321_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y4: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y4 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y4 != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [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))))
                     => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(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,Va))))
                     => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_4322_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),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,V)),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,V)),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,V)),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,V)),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_4323_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y4: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y4 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y4 != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y4 != 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))))
                     => ( Y4 = 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))))
                     => ( Y4 = 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_4324_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,V),TreeList,Vc2),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,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,V)),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,V)),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,V)),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,V)),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_4325_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y4) )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( pp(Y4)
            <=> ~ ( ( ( Xa = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B3) )
                      & ( 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(Y4) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)
               => ( pp(Y4)
                <=> ~ ( ( 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_4326_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa = zero_zero(nat) )
                 => pp(A4) )
                & ( ( Xa != zero_zero(nat) )
                 => ( ( ( Xa = one_one(nat) )
                     => pp(B3) )
                    & ( Xa = one_one(nat) ) ) ) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)
             => ~ ( ( 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_4327_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( ( Xa = zero_zero(nat) )
               => pp(A4) )
              & ( ( Xa != zero_zero(nat) )
               => ( ( ( Xa = one_one(nat) )
                   => pp(B3) )
                  & ( 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)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)
               => ( ( 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_4328_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)
             => ~ ( ( Xa = Mi2 )
                  | ( 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_4329_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_4330_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: 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),A2),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_am(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_4331_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_4332_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_ao(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_4333_predicate1I,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).

% predicate1I
tff(fact_4334_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_ap(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_4335_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,X))
     => ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
       => pp(aa(A,bool,Q,X)) ) ) ).

% rev_predicate1D
tff(fact_4336_predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),X: A] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,Q,X)) ) ) ).

% predicate1D
tff(fact_4337_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_4338_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_aq(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_4339_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_ar(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_4340_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_as(nat,real)) ).

% pi_series
tff(fact_4341_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_at(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_4342_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_4343_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_au(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_4344_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_av(nat,fun(complex,bool),N)),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_aw(complex,fun(nat,fun(complex,bool)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4345_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_at(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_4346_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_ax(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_4347_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_ay(nat,A)) ) ).

% summable_zero
tff(fact_4348_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_az(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_4349_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_ba(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_4350_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_ba(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_4351_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_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_4352_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_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_4353_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_bc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_4354_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bd(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_4355_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( summable(A,F2)
           => ( summable(A,G)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F2)),suminf(A,G))) ) ) ) ) ).

% suminf_le
tff(fact_4356_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_4357_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N5: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_4358_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_be(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_4359_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bd(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_4360_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_4361_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_az(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_4362_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_4363_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_ba(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_4364_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_4365_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_bg(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_bh(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_4366_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_4367_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) )
            <=> ! [N4: nat] : aa(nat,A,F2,N4) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_4368_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_4369_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bi(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_4370_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_4371_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_bk(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_4372_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_bm(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_4373_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_bn(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_4374_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_bo(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_4375_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F2,N2))),aa(nat,real,G,N2))) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_bp(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_4376_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bp(fun(nat,real),fun(nat,real),F2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_bp(fun(nat,real),fun(nat,real),F2)))) ) ).

% summable_rabs
tff(fact_4377_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_4378_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_4379_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bf(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_4380_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bq(fun(nat,A),fun(nat,real),F2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_bq(fun(nat,A),fun(nat,real),F2)))) ) ) ).

% summable_norm
tff(fact_4381_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N5: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N2: nat] : pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),aa(nat,complex,G,N2)),real_Vector_Reals(complex)))
           => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N2))))
             => ( ! [N2: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N2))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N2)))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_4382_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_bk(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_bk(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_4383_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_br(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_4384_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_bk(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bk(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_bl(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_4385_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_bk(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_bl(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_bk(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_4386_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
         => ( summable(A,F2)
           => ? [N9: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N7))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,fun(nat,A)),F2),N7)))),R2)) ) ) ) ) ).

% suminf_exist_split
tff(fact_4387_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_bt(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_4388_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M10: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R2),R0))
           => ( ! [N2: 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),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N2))),M10))
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bu(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_4389_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N5: 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),N5),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_4390_sin__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_bv(real,fun(nat,real),X),sin(real,X)) ).

% sin_paired
tff(fact_4391_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(aa(product_prod(vEBT_VEBT,nat),bool,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),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Uy2)),Xa))) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa)))
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
                   => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),Xa)))
                     => ( ( Xa = Mi2 )
                        | ( 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(product_prod(vEBT_VEBT,nat),bool,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_4392_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y4) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(Y4)
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ pp(Y4)
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Uy2)),Xa))) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( pp(Y4)
                    <=> ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa))) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
                   => ( ( pp(Y4)
                      <=> ( ( Xa = Mi2 )
                          | ( 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))) ) ) )
                     => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),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(Y4)
                        <=> ( ( 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))) ) )
                       => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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_4393_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_bw(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_4394_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_ay(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_4395_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),X: A] :
          ( sums(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),A2),X)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_4396_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% sums_diff
tff(fact_4397_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S2: A,T2: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( sums(A,F2,S2)
           => ( sums(A,G,T2)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S2),T2)) ) ) ) ) ).

% sums_le
tff(fact_4398_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_4399_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_ax(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_4400_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_bd(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_4401_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_bc(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_4402_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F2,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(A,fun(nat,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ).

% sums_divide
tff(fact_4403_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_bx(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_4404_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_by(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_4405_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_az(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_4406_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S2: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),F2),S2)
           => sums(A,F2,S2) ) ) ) ).

% sums_Suc_imp
tff(fact_4407_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S2: A] :
          ( sums(A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),F2),S2)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_4408_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_bz(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_4409_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F2: fun(nat,A),S2: 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_ca(nat,fun(fun(nat,A),fun(nat,A)),N),F2),S2)
          <=> sums(A,F2,S2) ) ) ) ).

% sums_zero_iff_shift
tff(fact_4410_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_cb(nat,fun(A,fun(nat,A)),M),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)) ) ).

% powser_sums_if
tff(fact_4411_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_4412_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_4413_power__half__series,axiom,
    sums(real,aTP_Lamp_cc(nat,real),one_one(real)) ).

% power_half_series
tff(fact_4414_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( sums(real,G,X)
     => sums(real,aTP_Lamp_cd(fun(nat,real),fun(nat,real),G),X) ) ).

% sums_if'
tff(fact_4415_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F2: fun(nat,real),Y4: real] :
      ( sums(real,G,X)
     => ( sums(real,F2,Y4)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ce(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y4)) ) ) ).

% sums_if
tff(fact_4416_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_cf(A,fun(nat,A),X),cosh(A,X)) ) ).

% cosh_converges
tff(fact_4417_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_cg(A,fun(nat,A),X),sinh(A,X)) ) ).

% sinh_converges
tff(fact_4418_cos__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_ch(real,fun(nat,real),X),cos(real,X)) ).

% cos_paired
tff(fact_4419_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa)))
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                ( ( X = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc)),Xa)))
                 => ~ ( ( Xa = Mi2 )
                      | ( 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(product_prod(vEBT_VEBT,nat),bool,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_4420_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A4,B3)),Xa)))
               => ( ( ( Xa = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B3) )
                      & ( 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(aa(product_prod(vEBT_VEBT,nat),bool,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))) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3) )
                 => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)),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_4421_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A4,B3)),Xa)))
               => ~ ( ( ( Xa = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B3) )
                        & ( Xa = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)),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_4422_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y4) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( pp(Y4)
                <=> ( ( ( Xa = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B3) )
                        & ( Xa = one_one(nat) ) ) ) ) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A4,B3)),Xa))) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ pp(Y4)
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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))) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3) )
                 => ( ( pp(Y4)
                    <=> ( ( 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))) ) )
                   => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S3)),Xa))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4423_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_ci(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_cj(fun(nat,A),fun(A,fun(nat,A)),C2),X),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_4424_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,C2),X4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X4))),aa(nat,A,C2,aa(nat,nat,suc,X4))) ) ).

% diffs_def
tff(fact_4425_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),X3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_4426_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K5: real,C2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),K5))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),K5))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cm(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_4427_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( topological_monoseq(A,X7)
        <=> ( ! [M5: nat,N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,M5)),aa(nat,A,X7,N4))) )
            | ! [M5: nat,N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N4)),aa(nat,A,X7,M5))) ) ) ) ) ).

% monoseq_def
tff(fact_4428_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [M4: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),aa(nat,A,X7,M4))) )
         => topological_monoseq(A,X7) ) ) ).

% monoI2
tff(fact_4429_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [M4: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,M4)),aa(nat,A,X7,N2))) )
         => topological_monoseq(A,X7) ) ) ).

% monoI1
tff(fact_4430_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( topological_monoseq(A,X7)
        <=> ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N4)),aa(nat,A,X7,aa(nat,nat,suc,N4))))
            | ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,aa(nat,nat,suc,N4))),aa(nat,A,X7,N4))) ) ) ) ).

% monoseq_Suc
tff(fact_4431_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),aa(nat,A,X7,aa(nat,nat,suc,N2))))
         => topological_monoseq(A,X7) ) ) ).

% mono_SucI1
tff(fact_4432_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,aa(nat,nat,suc,N2))),aa(nat,A,X7,N2)))
         => topological_monoseq(A,X7) ) ) ).

% mono_SucI2
tff(fact_4433_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_cn(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_4434_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero(complex) )
       => ( arg(Z) = zero_zero(real) ) )
      & ( ( Z != zero_zero(complex) )
       => ( arg(Z) = fChoice(real,aTP_Lamp_co(complex,fun(real,bool),Z)) ) ) ) ).

% Arg_def
tff(fact_4435_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_4436_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_cq(A,fun(A,fun(nat,A)),X),Y4),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y4))) ) ).

% sin_x_sin_y
tff(fact_4437_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),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),A3)),ring_1_Ints(B))) ) ) ).

% Ints_sum
tff(fact_4438_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cr(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_4439_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_4440_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B4: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,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),A2),set_ord_atMost(nat,N2))),B4))
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_4441_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ct(nat,fun(nat,nat),R2)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),N))),N) ).

% sum_choose_lower
tff(fact_4442_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D2: fun(nat,A)] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cu(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),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_cu(fun(nat,A),fun(A,fun(nat,A)),D2),X5)),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_4443_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: set(B)] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),aTP_Lamp_cv(fun(B,A),fun(B,real),F2)),A3))) ) ).

% norm_sum
tff(fact_4444_sum__norm__le,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S: set(B),F2: fun(B,A),G: fun(B,real)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(B,A,F2,X3))),aa(B,real,G,X3))) )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),G),S))) ) ) ).

% sum_norm_le
tff(fact_4445_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_cw(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3),A2) ) ).

% mod_sum_eq
tff(fact_4446_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,bool),A3: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,P)))
      <=> pp(A3) )
     => ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
      <=> pp(A3) ) ) ).

% verit_sko_ex'
tff(fact_4447_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X_12: A] : pp(aa(A,bool,P,X_12))
    <=> pp(aa(A,bool,P,fChoice(A,aTP_Lamp_ae(fun(A,bool),fun(A,bool),P)))) ) ).

% verit_sko_forall
tff(fact_4448_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,bool),A3: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,aTP_Lamp_ae(fun(A,bool),fun(A,bool),P))))
      <=> pp(A3) )
     => ( ! [X_12: A] : pp(aa(A,bool,P,X_12))
      <=> pp(A3) ) ) ).

% verit_sko_forall'
tff(fact_4449_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B4: A,A3: A,P: fun(A,bool)] :
      ( ( B4 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_4450_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_ex_indirect
tff(fact_4451_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P2: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
          <=> pp(aa(A,bool,P2,X3)) )
       => ( ? [X_12: A] : pp(aa(A,bool,P2,X_12))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_4452_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_ae(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X_12: A] : pp(aa(A,bool,P,X_12))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_forall_indirect
tff(fact_4453_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P2: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_ae(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
          <=> pp(aa(A,bool,P2,X3)) )
       => ( ! [X_12: A] : pp(aa(A,bool,P2,X_12))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_4454_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cx(nat,fun(nat,nat),M)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_4455_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_cy(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_4456_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cu(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),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_4457_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cz(A,fun(nat,A),A2)),set_ord_atMost(nat,N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_4458_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_da(nat,fun(nat,fun(nat,nat)),M),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_4459_vandermonde,axiom,
    ! [M: nat,N: nat,R2: 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_db(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R2) ).

% vandermonde
tff(fact_4460_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% choose_row_sum
tff(fact_4461_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),N) = aa(set(nat),nat,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_dc(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),N)),set_ord_atMost(nat,N)) ).

% binomial
tff(fact_4462_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4463_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4464_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: 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),M),I3))
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(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_di(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ) ).

% polynomial_product
tff(fact_4465_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dj(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4466_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N) = aa(set(nat),A,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_dk(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),set_ord_atMost(nat,N)) ) ).

% binomial_ring
tff(fact_4467_polynomial__product__nat,axiom,
    ! [M: nat,A2: 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),M),I3))
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dl(fun(nat,nat),fun(nat,fun(nat,nat)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dl(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_dn(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_4468_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_do(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),N) ).

% choose_square_sum
tff(fact_4469_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),N) = aa(set(nat),A,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_dp(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),set_ord_atMost(nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_4470_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_df(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4471_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_4472_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_4473_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y4: 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_dq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y4)),set_ord_atMost(nat,M)) = 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_dr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y4)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4474_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y4: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4) = aa(A,A,aa(A,fun(A,A),times_times(A),Y4),X) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),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_ds(A,fun(A,fun(nat,fun(nat,A))),X),Y4),N)),set_ord_atMost(nat,N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4475_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A2: 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) = A2 )
          <=> ( 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_dt(nat,fun(A,fun(A,fun(nat,A))),N),Z),A2)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4476_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_du(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4477_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dv(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4478_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y4: 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_dq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y4)),set_ord_atMost(nat,M)) = 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_dw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y4)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4479_binomial__r__part__sum,axiom,
    ! [M: 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))),M)),one_one(nat)))),set_ord_atMost(nat,M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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))),M)) ).

% binomial_r_part_sum
tff(fact_4480_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dx(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_4481_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_dy(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4482_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))
         => ? [M8: real] :
            ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z3)))
             => 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_bg(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4483_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: 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),M))),one_one(A)))),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_4484_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_dz(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_4485_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_ea(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_4486_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eb(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4487_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_ec(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_4488_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_ec(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_4489_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ee(A,fun(A,fun(nat,A)),X),Y4),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y4))) ) ).

% cos_x_cos_y
tff(fact_4490_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y4: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_eg(A,fun(A,fun(nat,A)),X),Y4),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4))) ) ).

% sums_cos_x_plus_y
tff(fact_4491_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: 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_eh(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs_ge_zero
tff(fact_4492_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_4493_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_4494_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: 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)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
           => ( 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_ei(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P3)) = 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_ej(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),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4495_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atMost(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),K)) ) ) ).

% atMost_iff
tff(fact_4496_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ek(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_4497_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,X)),set_ord_atMost(A,Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% atMost_subset_iff
tff(fact_4498_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_eh(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs
tff(fact_4499_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X3)),aa(A,nat,F2,X3))) )
     => ( 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_el(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_4500_int__sum,axiom,
    ! [B: $tType,F2: fun(B,nat),A3: 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),A3)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_em(fun(B,nat),fun(B,int),F2)),A3) ).

% int_sum
tff(fact_4501_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S2: set(A)] : aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7311177749621191930dd_sum(A,complex),aTP_Lamp_en(fun(A,real),fun(A,complex),F2)),S2) = complex2(aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),F2),S2),zero_zero(real)) ).

% Complex_sum'
tff(fact_4502_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,N) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X3))) ) ) ).

% sum_SucD
tff(fact_4503_bounded__Max__nat,axiom,
    ! [P: fun(nat,bool),X: nat,M10: nat] :
      ( pp(aa(nat,bool,P,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M10)) )
       => ~ ! [M4: nat] :
              ( pp(aa(nat,bool,P,M4))
             => ~ ! [X4: nat] :
                    ( pp(aa(nat,bool,P,X4))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M4)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_4504_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_eo(complex,complex)),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_ep(nat,fun(complex,fun(complex,bool)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4505_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_eo(complex,complex)),collect(complex,aTP_Lamp_av(nat,fun(complex,bool),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4506_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4507_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,G,X3) = zero_zero(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_4508_dvd__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),D2: A,F2: fun(B,A)] :
          ( ! [A4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(B,A,F2,A4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))) ) ) ).

% dvd_sum
tff(fact_4509_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: 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),K5))
             => 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),groups7311177749621191930dd_sum(B,A),F2),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),K5))) ) ) ).

% sum_mono
tff(fact_4510_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F2: fun(A,B),A3: set(A),G: fun(C,B),B4: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_er(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F2),G),B4)),A3) ) ).

% sum_product
tff(fact_4511_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_es(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_distrib_right
tff(fact_4512_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) = 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_et(A,fun(fun(B,A),fun(B,A)),R2),F2)),A3) ) ).

% sum_distrib_left
tff(fact_4513_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_eu(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ).

% sum_subtractf
tff(fact_4514_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_ev(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_divide_distrib
tff(fact_4515_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_ew(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_4516_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),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),A3)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_4517_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X3))) )
         => 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),A3))) ) ) ).

% sum_nonneg
tff(fact_4518_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
         => ( ! [X3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X3)),A3))
               => ( aa(nat,A,F2,aa(nat,nat,suc,X3)) = aa(nat,A,G,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_4519_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ex(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),I5)) ) ).

% sum_power_add
tff(fact_4520_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_ey(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_4521_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_ez(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_4522_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [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,A2,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_fa(fun(A,B),fun(fun(A,B),fun(A,B)),X),A2)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4523_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_4524_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_fb(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.in_pairs_0
tff(fact_4525_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)))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),zero_zero(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_fc(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4526_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))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),X))
            & ( 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_fc(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4527_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))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),X))
            & ( 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_fd(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4528_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),X))
          & ( 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_fd(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4529_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_lessThan(A,X)),set_ord_lessThan(A,Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% lessThan_subset_iff
tff(fact_4530_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_4531_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_fe(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4532_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_4533_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_4534_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_ff(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_4535_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3)))
         => ( 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_fg(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).

% sum_diff_distrib
tff(fact_4536_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,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))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F2)),X)) ) ) ) ).

% suminf_le_const
tff(fact_4537_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_ey(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_4538_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),N: nat,R2: 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)),R2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),R2)),set_ord_lessThan(nat,N)) ) ).

% sumr_diff_mult_const2
tff(fact_4539_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ez(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_4540_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_4541_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_4542_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),N: nat,S2: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,fun(nat,A)),F2),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),S2),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,S2) ) ) ).

% sums_iff_shift'
tff(fact_4543_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S2: A,N: nat] :
          ( sums(A,F2,S2)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,fun(nat,A)),F2),N),aa(A,A,aa(A,fun(A,A),minus_minus(A),S2),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_4544_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_4545_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_4546_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_4547_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_ey(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.atMost_shift
tff(fact_4548_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_bs(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_4549_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,M4))) )
           => 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_4550_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_4551_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: 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_fj(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) = 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_fk(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_4552_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y4: 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),Y4),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)),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_fl(A,fun(nat,fun(A,fun(nat,A))),X),N),Y4)),set_ord_lessThan(nat,N))) ) ).

% power_diff_sumr2
tff(fact_4553_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y4: 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),Y4),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),Y4)),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_fm(A,fun(nat,fun(A,fun(nat,A))),X),N),Y4)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_4554_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: 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_dg(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,N)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),set_ord_lessThan(nat,N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4555_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),N: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),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_dg(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,N))) ) ).

% polyfun_linear_factor
tff(fact_4556_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F2: fun(nat,A),K5: 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)),K5)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),K5))
           => 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)),K5))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_4557_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M4))) )
           => ( 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_4558_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_fn(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq'
tff(fact_4559_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_fo(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_4560_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B8: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_fp(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N)),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_4561_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_fq(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_fr(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_fs(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,N))) ).

% sum_split_even_odd
tff(fact_4562_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),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_ft(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_4563_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y4: 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_dg(fun(nat,A),fun(A,fun(nat,A)),A2),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_dg(fun(nat,A),fun(A,fun(nat,A)),A2),Y4)),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),Y4)),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_fv(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y4)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4564_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),sin(real,X)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fd(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_4565_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D3: 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)))),D3)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat)))))))
       => 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_4566_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))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4)))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T4)),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_ft(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4567_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_fx(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_4568_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: 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_fd(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4569_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
      & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fd(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4570_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
      & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fc(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),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_4571_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_fy(nat,fun(nat,complex),N),set_ord_lessThan(nat,N),collect(complex,aTP_Lamp_av(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_4572_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( 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,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( ( ( 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,M,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))),M)) ) )
              & ( ( 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,M,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),M)),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_4573_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eb(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4574_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_4575_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_4576_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L4: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L4,H2) )
        <=> ( ( ( L = L4 )
              & ( H = H2 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),H2)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_4577_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: 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,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4578_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H2)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_4579_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,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)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_4580_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L4: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,H)),set_or1337092689740270186AtMost(A,L4,H2))) ) ).

% not_Iic_le_Icc
tff(fact_4581_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)) )
    <=> ! [X5: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X5)) ) ) ).

% all_nat_less
tff(fact_4582_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)) )
    <=> ? [X5: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X5)) ) ) ).

% ex_nat_less
tff(fact_4583_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_4584_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4585_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: 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,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
                & 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),A2))
                  | 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_4586_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = 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_fz(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4587_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_4588_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_4589_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_4590_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_4591_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4592_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,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,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_4593_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_ey(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4594_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_ey(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_4595_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: 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_ga(fun(nat,fun(nat,A)),fun(nat,A),A2)),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_gc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_ord_lessThan(nat,N)) ) ).

% sum.nested_swap'
tff(fact_4596_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gd(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4597_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = 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,M,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),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4598_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: 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),M),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,M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ).

% sum_up_index_split
tff(fact_4599_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ge(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ge(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_4600_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gf(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_4601_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,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),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4602_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E2: real] :
          ( summable(A,F2)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ~ ! [N9: nat] :
                  ~ ! [M2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),M2))
                     => ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,M2,N7)))),E2)) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4603_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_4604_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: 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))),M),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_fb(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_4605_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cu(fun(nat,A),fun(A,fun(nat,A)),C2),X5)),set_ord_atMost(nat,N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X5: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C2,X5) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4606_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_gg(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_4607_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(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_4608_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: 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_gi(A,fun(A,fun(nat,A)),A2),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))),A2)),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_4609_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_4610_arith__series__nat,axiom,
    ! [A2: 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_gj(nat,fun(nat,fun(nat,nat)),A2),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))),A2)),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_4611_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(nat,nat)),set_or1337092689740270186AtMost(nat,M,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),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_4612_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_4613_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: 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_gk(A,fun(A,fun(nat,A)),A2),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))),A2)),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_4614_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_4615_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,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_4616_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y4: 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_dg(fun(nat,A),fun(A,fun(nat,A)),A2),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_dg(fun(nat,A),fun(A,fun(nat,A)),A2),Y4)),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),Y4)),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_gm(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y4)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff
tff(fact_4617_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))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_gn(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_4618_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_4619_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),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_go(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4620_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),ring_1_Ints(B))) )
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),groups7121269368397514597t_prod(A,B,F2,A3)),ring_1_Ints(B))) ) ) ).

% Ints_prod
tff(fact_4621_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gp(fun(B,nat),fun(B,A),F2),A3) ) ).

% of_nat_prod
tff(fact_4622_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gq(fun(B,int),fun(B,A),F2),A3) ) ).

% of_int_prod
tff(fact_4623_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_4624_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_4625_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_4626_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),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_gr(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4627_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gs(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_4628_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gt(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,H,A3)) ) ).

% prod.distrib
tff(fact_4629_prod__dvd__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [A4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,A4)),aa(B,A,G,A4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ).

% prod_dvd_prod
tff(fact_4630_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A3: set(B)] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A3))),groups7121269368397514597t_prod(B,real,aTP_Lamp_gu(fun(B,A),fun(B,real),F2),A3))) ) ).

% norm_prod_le
tff(fact_4631_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_cw(fun(B,A),fun(A,fun(B,A)),F2),A2),A3),A2) = modulo_modulo(A,groups7121269368397514597t_prod(B,A,F2,A3),A2) ) ).

% mod_prod_eq
tff(fact_4632_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: 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),A3))
             => ( 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),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ).

% prod_mono
tff(fact_4633_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_nonneg
tff(fact_4634_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_pos
tff(fact_4635_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_ge_1
tff(fact_4636_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4637_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7121269368397514597t_prod(B,A,F2,A3)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_4638_bset_I1_J,axiom,
    ! [D4: int,B4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                & pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)))
                & pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4))) ) ) ) ) ) ).

% bset(1)
tff(fact_4639_bset_I2_J,axiom,
    ! [D4: int,B4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                | pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)))
                | pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4))) ) ) ) ) ) ).

% bset(2)
tff(fact_4640_aset_I1_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                & pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)))
                & pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ) ) ).

% aset(1)
tff(fact_4641_aset_I2_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                | pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)))
                | pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ) ) ).

% aset(2)
tff(fact_4642_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gw(fun(nat,A),fun(nat,fun(nat,A)),G),N),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_4643_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4644_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_4645_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4646_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4647_bset_I9_J,axiom,
    ! [D2: int,D4: int,B4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),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),X4),D4)),T2))) ) ) ) ).

% bset(9)
tff(fact_4648_bset_I10_J,axiom,
    ! [D2: int,D4: int,B4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),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),X4),D4)),T2))) ) ) ) ).

% bset(10)
tff(fact_4649_aset_I9_J,axiom,
    ! [D2: int,D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),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),X4),D4)),T2))) ) ) ) ).

% aset(9)
tff(fact_4650_aset_I10_J,axiom,
    ! [D2: int,D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),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),X4),D4)),T2))) ) ) ) ).

% aset(10)
tff(fact_4651_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4652_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4653_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_4654_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4655_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_gh(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_4656_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_gy(fun(nat,fun(nat,A)),fun(nat,A),A2),set_ord_atMost(nat,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ha(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N),set_ord_lessThan(nat,N)) ) ).

% prod.nested_swap'
tff(fact_4657_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_hb(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4658_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_4659_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))
     => ( ! [X3: int,K2: 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),K2),D2)))) )
       => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
        <=> ? [X5: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),set_or1337092689740270186AtMost(int,one_one(int),D2)))
              & pp(aa(int,bool,P,X5)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4660_bset_I3_J,axiom,
    ! [D4: int,T2: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( X4 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_4661_bset_I4_J,axiom,
    ! [D4: int,T2: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( X4 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_4662_bset_I5_J,axiom,
    ! [D4: int,B4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X4),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)),T2)) ) ) ) ).

% bset(5)
tff(fact_4663_bset_I7_J,axiom,
    ! [D4: int,T2: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X4))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4))) ) ) ) ) ).

% bset(7)
tff(fact_4664_aset_I3_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( ( X4 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_4665_aset_I4_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( ( X4 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_4666_aset_I5_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X4),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),T2)) ) ) ) ) ).

% aset(5)
tff(fact_4667_aset_I7_J,axiom,
    ! [D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X4))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ).

% aset(7)
tff(fact_4668_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),groups7121269368397514597t_prod(I6,A,Z,I5)),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_hc(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Z),W)),I5))) ) ) ) ).

% norm_prod_diff
tff(fact_4669_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.atMost_shift
tff(fact_4670_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_gh(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4671_bset_I6_J,axiom,
    ! [D4: int,B4: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X4),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),X4),D4)),T2)) ) ) ) ).

% bset(6)
tff(fact_4672_bset_I8_J,axiom,
    ! [D4: int,T2: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X4))
             => 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),X4),D4))) ) ) ) ) ).

% bset(8)
tff(fact_4673_aset_I6_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X4),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),X4),D4)),T2)) ) ) ) ) ).

% aset(6)
tff(fact_4674_aset_I8_J,axiom,
    ! [D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X4))
           => 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),X4),D4))) ) ) ) ).

% aset(8)
tff(fact_4675_cppi,axiom,
    ! [D4: int,P: fun(int,bool),P2: fun(int,bool),A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z3),X3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P2,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
            <=> ( ? [X5: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P2,X5)) )
                | ? [X5: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),A3))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa4),X5))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4676_cpmi,axiom,
    ! [D4: int,P: fun(int,bool),P2: fun(int,bool),B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P2,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B4))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
            <=> ( ? [X5: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P2,X5)) )
                | ? [X5: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),B4))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X5))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4677_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hd(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_4678_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_he(A,fun(nat,fun(nat,A)),A2),N),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_4679_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_gh(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4680_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),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)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hf(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_4681_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hf(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.in_pairs_0
tff(fact_4682_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_he(A,fun(nat,fun(nat,A)),A2),N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4683_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: 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)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hg(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,P3)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hh(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),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4684_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hi(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4685_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_hj(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_4686_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_hk(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_4687_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N))
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_hl(int,int)),set_or1337092689740270186AtMost(int,M,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),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_4688_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_hm(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4689_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hn(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_4690_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y4: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y4 )
     => ( pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y4 = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat))) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y4 = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa(nat,bool,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))))
                       => ( Y4 = 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))))
                       => ( Y4 = 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)))))) ) ) )
                   => ~ pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4691_arctan__def,axiom,
    ! [Y4: real] : aa(real,real,arctan,Y4) = the(real,aTP_Lamp_ho(real,fun(real,bool),Y4)) ).

% arctan_def
tff(fact_4692_int__prod,axiom,
    ! [B: $tType,F2: fun(B,nat),A3: set(B)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,int,aTP_Lamp_em(fun(B,nat),fun(B,int),F2),A3) ).

% int_prod
tff(fact_4693_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hp(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_4694_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hs(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.triangle_reindex
tff(fact_4695_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,J)) = groups7121269368397514597t_prod(int,int,aTP_Lamp_hl(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_4696_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_hp(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_hu(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_4697_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_hs(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_hu(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.triangle_reindex
tff(fact_4698_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_hl(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_4699_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_hv(real,bool)) ) ) ).

% ln_neg_is_const
tff(fact_4700_arccos__def,axiom,
    ! [Y4: real] : aa(real,real,arccos,Y4) = the(real,aTP_Lamp_hw(real,fun(real,bool),Y4)) ).

% arccos_def
tff(fact_4701_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_4702_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_hx(real,bool)) ).

% pi_half
tff(fact_4703_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_hx(real,bool))) ).

% pi_def
tff(fact_4704_arcsin__def,axiom,
    ! [Y4: real] : aa(real,real,arcsin,Y4) = the(real,aTP_Lamp_hy(real,fun(real,bool),Y4)) ).

% arcsin_def
tff(fact_4705_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_hz(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_4706_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y4: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y4 )
     => ( pp(aa(nat,bool,accp(nat,vEBT_V1247956027447740395_p_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ pp(aa(nat,bool,accp(nat,vEBT_V1247956027447740395_p_rel),zero_zero(nat))) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ pp(aa(nat,bool,accp(nat,vEBT_V1247956027447740395_p_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))))
                       => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(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,Va))))
                       => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(nat)))) ) ) )
                   => ~ pp(aa(nat,bool,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_4707_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y4: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y4 )
     => ( pp(aa(nat,bool,accp(nat,vEBT_V5144397997797733112_d_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) )
             => ~ pp(aa(nat,bool,accp(nat,vEBT_V5144397997797733112_d_rel),zero_zero(nat))) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y4 = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))) )
               => ~ pp(aa(nat,bool,accp(nat,vEBT_V5144397997797733112_d_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))))
                       => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),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))))
                       => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),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_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) )
                   => ~ pp(aa(nat,bool,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_4708_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_ia(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_4709_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),R)),aTP_Lamp_a(set(A),fun(A,bool),S)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R),S)) ) ).

% pred_subset_eq
tff(fact_4710_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_4711_set__decode__inverse,axiom,
    ! [N: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(N)) = N ).

% set_decode_inverse
tff(fact_4712_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Y: 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,X3),Y)),R2))
         => 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,X3),Y)),S2)) )
     => 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))),R2),S2)) ) ).

% subrelI
tff(fact_4713_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aTP_Lamp_ib(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),aTP_Lamp_ib(set(product_prod(A,B)),fun(A,fun(B,bool)),S)))
    <=> 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))),R),S)) ) ).

% pred_subset_eq2
tff(fact_4714_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_4715_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(nat,nat)),set_or7035219750837199246ssThan(nat,M,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),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_4716_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_4717_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ! [X3: A,Y: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Y))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Y)) )
     => 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))),P),Q)) ) ).

% predicate2I
tff(fact_4718_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_4719_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),I))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),N)) ) ) ) ) ).

% ivl_subset
tff(fact_4720_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,N: A,M: 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,M)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_4721_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: A,M: 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,M)) = set_or7035219750837199246ssThan(A,M,N) ) ).

% lessThan_minus_lessThan
tff(fact_4722_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_4723_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_4724_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Y4: B,Q: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y4))
     => ( 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))),P),Q))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y4)) ) ) ).

% rev_predicate2D
tff(fact_4725_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),X: A,Y4: B] :
      ( 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))),P),Q))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y4))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y4)) ) ) ).

% predicate2D
tff(fact_4726_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: 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,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4727_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)) )
    <=> ? [X5: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_4728_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)) )
    <=> ! [X5: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X5)) ) ) ).

% all_nat_less_eq
tff(fact_4729_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_4730_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_4731_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: 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,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4732_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4733_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,A2,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_4734_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( groups7121269368397514597t_prod(B,A,G,set_or7035219750837199246ssThan(B,A2,B2)) = groups7121269368397514597t_prod(B,A,H,set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4735_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P3: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( 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,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4736_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P3: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( 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,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,N,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4737_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y4: 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_eq(nat),Y4),aa(A,nat,F2,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),size_list(A,F2,Xs))) ) ) ).

% size_list_estimation'
tff(fact_4738_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X3)),aa(A,nat,G,X3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),size_list(A,F2,Xs)),size_list(A,G,Xs))) ) ).

% size_list_pointwise
tff(fact_4739_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P3: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,P3))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4740_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4741_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: 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,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4742_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_4743_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_4744_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4745_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4746_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4747_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4748_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4749_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_4750_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_4751_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4752_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = 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_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4753_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: 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_id(fun(nat,fun(nat,A)),fun(nat,A),A2)),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_gc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_4754_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ie(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4755_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ha(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_4756_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,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_ig(fun(nat,A),fun(nat,fun(nat,A)),G),K)),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,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_4757_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ih(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_4758_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_4759_prod__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_4760_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_4761_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_4762_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_4763_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = 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_fz(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4764_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4765_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hd(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_4766_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_4767_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [N6: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),M5))
                 => ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M5,N4)))),E3)) ) ) ) ) ).

% summable_Cauchy
tff(fact_4768_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S2: A,K: nat] :
          ( sums(A,F2,S2)
         => ( 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_ii(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S2) ) ) ) ).

% sums_group
tff(fact_4769_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ij(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_4770_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),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_4771_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)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ik(nat,fun(nat,fun(nat,A)),K),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4772_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_il(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4773_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_im(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4774_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_im(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4775_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hi(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4776_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: 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),A2),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_in(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4777_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,I3)),aa(nat,A,A2,J2))) ) )
         => ( ! [I3: nat,J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,B2,J2)),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_io(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),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_4778_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,A2,I3)),aa(nat,nat,A2,J2))) ) )
     => ( ! [I3: nat,J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,B2,J2)),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_ip(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),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_4779_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_4780_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_4781_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,bool)),R22: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R1),R22))
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),accp(A,R22)),accp(A,R1))) ) ).

% accp_subset
tff(fact_4782_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,bool)),B4: 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))),A3),B4))
     => 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))),collect(product_prod(A,B),product_case_prod(A,B,bool,A3))),collect(product_prod(A,B),product_case_prod(A,B,bool,B4)))) ) ).

% Collect_case_prod_mono
tff(fact_4783_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_4784_accp__subset__induct,axiom,
    ! [A: $tType,D4: fun(A,bool),R: fun(A,fun(A,bool)),X: A,P: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),D4),accp(A,R)))
     => ( ! [X3: A,Z2: A] :
            ( pp(aa(A,bool,D4,X3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R,Z2),X3))
             => pp(aa(A,bool,D4,Z2)) ) )
       => ( pp(aa(A,bool,D4,X))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,D4,X3))
               => ( ! [Z3: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),R,Z3),X3))
                     => pp(aa(A,bool,P,Z3)) )
                 => pp(aa(A,bool,P,X3)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_4785_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y4: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y4 )
     => ( pp(aa(product_prod(nat,nat),bool,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))
               => ( Y4 = 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))
               => ( Y4 = 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))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,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_4786_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y4: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y4 )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y4 = one2 )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [M4: num] :
                  ( ( Xa = aa(num,num,bit0,M4) )
                 => ( ( Y4 = aa(num,num,bit1,M4) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,M4)))) ) ) )
           => ( ( ( X = one2 )
               => ! [M4: num] :
                    ( ( Xa = aa(num,num,bit1,M4) )
                   => ( ( Y4 = aa(num,num,bit1,M4) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,M4)))) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa(num,num,bit0,N2) )
                   => ( ( Xa = one2 )
                     => ( ( Y4 = aa(num,num,bit0,one2) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),one2))) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa(num,num,bit0,N2) )
                     => ! [M4: num] :
                          ( ( Xa = aa(num,num,bit0,M4) )
                         => ( ( Y4 = bitM(bit_or_not_num_neg(N2,M4)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),aa(num,num,bit0,M4)))) ) ) )
                 => ( ! [N2: num] :
                        ( ( X = aa(num,num,bit0,N2) )
                       => ! [M4: num] :
                            ( ( Xa = aa(num,num,bit1,M4) )
                           => ( ( Y4 = aa(num,num,bit0,bit_or_not_num_neg(N2,M4)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),aa(num,num,bit1,M4)))) ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa(num,num,bit1,N2) )
                         => ( ( Xa = one2 )
                           => ( ( Y4 = one2 )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),one2))) ) ) )
                     => ( ! [N2: num] :
                            ( ( X = aa(num,num,bit1,N2) )
                           => ! [M4: num] :
                                ( ( Xa = aa(num,num,bit0,M4) )
                               => ( ( Y4 = bitM(bit_or_not_num_neg(N2,M4)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),aa(num,num,bit0,M4)))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( X = aa(num,num,bit1,N2) )
                             => ! [M4: num] :
                                  ( ( Xa = aa(num,num,bit1,M4) )
                                 => ( ( Y4 = bitM(bit_or_not_num_neg(N2,M4)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),aa(num,num,bit1,M4)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_4787_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_iq(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than_def
tff(fact_4788_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_ir(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than2_def
tff(fact_4789_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,A0),A1)))
     => ( ! [I3: int,J2: int] :
            ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I3),J2)))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J2))
               => 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))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_4790_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral(int) ).

% Code_Target_Int.positive_def
tff(fact_4791_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_is(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4792_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V: num,N: nat] : aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)),aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N))) ).

% rec_nat_add_eq_if
tff(fact_4793_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num,N: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_4794_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_4795_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_4796_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num] : case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F2,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_4797_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V: num] : aa(nat,A,rec_nat(A,A2,F2),aa(num,nat,numeral_numeral(nat),V)) = aa(A,A,aa(nat,fun(A,A),F2,pred_numeral(V)),aa(nat,A,rec_nat(A,A2,F2),pred_numeral(V))) ).

% rec_nat_numeral
tff(fact_4798_less__eq__integer__code_I1_J,axiom,
    pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_eq_integer_code(1)
tff(fact_4799_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_it(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_4800_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_4801_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_4802_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_4803_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_4804_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_4805_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_4806_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_4807_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_4808_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_4809_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_iu(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_4810_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_iv(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_4811_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_4812_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_4813_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_4814_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_iw(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_4815_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ix(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_4816_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_gh(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).

% diff_Suc
tff(fact_4817_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),N))
        <=> pp(case_nat(bool,fFalse,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(1)
tff(fact_4818_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),N))
        <=> pp(case_nat(bool,fTrue,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(2)
tff(fact_4819_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_4820_old_Orec__nat__def,axiom,
    ! [T: $tType,X4: T,Xa2: fun(nat,fun(T,T)),Xb: nat] : aa(nat,T,rec_nat(T,X4,Xa2),Xb) = the(T,rec_set_nat(T,X4,Xa2,Xb)) ).

% old.rec_nat_def
tff(fact_4821_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( code_integer_of_int(K) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( code_integer_of_int(K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( code_integer_of_int(K) = if(code_integer,aa(int,bool,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))),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))),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_4822_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_4823_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa),X)) ).

% flip_bit_integer.abs_eq
tff(fact_4824_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_4825_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa),X)) ).

% unset_bit_integer.abs_eq
tff(fact_4826_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),code_integer_of_int(Xa)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),X)) ).

% divide_integer.abs_eq
tff(fact_4827_set__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa),X)) ).

% set_bit_integer.abs_eq
tff(fact_4828_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_4829_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_4830_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_4831_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xa),X)) ).

% times_integer.abs_eq
tff(fact_4832_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_4833_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),code_integer_of_int(Xa)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),X)) ).

% minus_integer.abs_eq
tff(fact_4834_subset__CollectI,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),Q: fun(A,bool),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
           => ( pp(aa(A,bool,Q,X3))
             => pp(aa(A,bool,P,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),B4),Q))),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),A3),P)))) ) ) ).

% subset_CollectI
tff(fact_4835_subset__Collect__iff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))))
      <=> ! [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4))
           => pp(aa(A,bool,P,X5)) ) ) ) ).

% subset_Collect_iff
tff(fact_4836_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_iy(real,fun(int,bool),X)) ).

% floor_real_def
tff(fact_4837_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_4838_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_iz(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_4839_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_4840_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_4841_less__eq__rat__def,axiom,
    ! [X: rat,Y4: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y4))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y4))
        | ( X = Y4 ) ) ) ).

% less_eq_rat_def
tff(fact_4842_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
     => ~ ! [S3: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S3))
           => ! [T4: rat] :
                ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T4))
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4843_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero(rat) )
       => ( aa(rat,rat,sgn_sgn(rat),A2) = zero_zero(rat) ) )
      & ( ( A2 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( aa(rat,rat,sgn_sgn(rat),A2) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( aa(rat,rat,sgn_sgn(rat),A2) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_4844_abs__rat__def,axiom,
    ! [A2: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = aa(rat,rat,uminus_uminus(rat),A2) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = A2 ) ) ) ).

% abs_rat_def
tff(fact_4845_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_4846_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit0,N)) = 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)) ).

% integer_of_num(2)
tff(fact_4847_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa(num,num,bit0,one2)) = aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)) ).

% integer_of_num_triv(2)
tff(fact_4848_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_gh(nat,nat),Nat) ).

% pred_def
tff(fact_4849_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_4850_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ja(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_4851_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q2),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_4852_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_4853_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_4854_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),zero_zero(int))) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_4855_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_4856_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_4857_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q2),R2) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q2),aa(rat,rat,uminus_uminus(rat),R2)) ).

% diff_rat_def
tff(fact_4858_rat__times__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_jc(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P3)) ).

% rat_times_code
tff(fact_4859_rat__divide__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_je(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_4860_rat__plus__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_jg(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_4861_rat__minus__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ji(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_4862_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_4863_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2)) ) ).

% normalize_denom_pos
tff(fact_4864_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P3: int,R2: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,R2),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4865_rat__less__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P3),Q2))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_jk(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_4866_rat__less__eq__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P3),Q2))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_jm(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_4867_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_jn(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_4868_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_4869_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: set(nat),C2: fun(nat,A),D2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( 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_jo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( 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_jo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_4870_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_4871_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),F3))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),F3) = zero_zero(A) )
          <=> ! [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),F3))
               => ( aa(B,A,F2,X5) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4872_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( groups7121269368397514597t_prod(B,A,F2,A3) = zero_zero(A) )
          <=> ? [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
                & ( aa(B,A,F2,X5) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_4873_dvd__prod__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,B2: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => ( ( B2 = aa(B,A,F2,A2) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_4874_dvd__prodI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,A2)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ) ).

% dvd_prodI
tff(fact_4875_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( 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_jp(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_jp(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_4876_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( 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_jq(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_jq(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_4877_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_jr(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_4878_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),A3))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_js(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_4879_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_4880_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F2,A3)))
      <=> ! [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X5))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4881_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jt(fun(nat,A),fun(nat,A),C2)),A3) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jt(fun(nat,A),fun(nat,A),C2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_4882_finite__set__decode,axiom,
    ! [N: nat] : pp(aa(set(nat),bool,finite_finite(nat),nat_set_decode(N))) ).

% finite_set_decode
tff(fact_4883_set__encode__eq,axiom,
    ! [A3: set(nat),B4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( pp(aa(set(nat),bool,finite_finite(nat),B4))
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B4) )
        <=> ( A3 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_4884_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),N5))
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),N5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),M5)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_4885_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ju(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_eq
tff(fact_4886_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,F2,N2)))
     => pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_jv(fun(nat,nat),fun(nat,fun(nat,bool)),F2),U)))) ) ).

% finite_less_ub
tff(fact_4887_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(I6,A),I5: set(I6),G: fun(I6,A),I: I6] :
          ( ( aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F2),I5) = aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G),I5) )
         => ( ! [I3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F2,I3)),aa(I6,A,G,I3))) )
           => ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I),I5))
             => ( pp(aa(set(I6),bool,finite_finite(I6),I5))
               => ( aa(I6,A,F2,I) = aa(I6,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4888_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                & ( aa(B,A,F2,X4) = zero_zero(A) ) )
           => ( groups7121269368397514597t_prod(B,A,F2,A3) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_4889_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_jw(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_le
tff(fact_4890_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N5))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N5))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_4891_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y4: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jx(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_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),Y4))))
           => 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_jy(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y4)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_4892_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y4: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jz(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_jz(set(B),fun(fun(B,A),fun(B,bool)),I5),Y4))))
           => 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_ka(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y4)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_4893_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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_kb(set(B),fun(fun(B,bool),fun(B,bool)),A3),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_kc(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_4894_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_4895_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_kd(A,fun(A,fun(A,bool)),A2),B2)))) ) ).

% finite_int_segment
tff(fact_4896_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: 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),S2))
         => ( pp(aa(set(C),bool,finite_finite(C),T2))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ? [Xa2: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa2),T2))
                        & ( aa(C,B,I,Xa2) = X3 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),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),S2)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_4897_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3) = zero_zero(A) )
            <=> ! [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
                 => ( aa(B,A,F2,X5) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4898_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: set(I6),F2: fun(I6,A),G: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite(I6),A3))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F2,X3)),aa(I6,A,G,X3))) )
           => ( ? [X4: I6] :
                  ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F2,X4)),aa(I6,A,G,X4))) )
             => 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),A3)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G),A3))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4899_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,zero_zero(A)),zero_zero(A)))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S))) ) ) ) ) ) ).

% sum.related
tff(fact_4900_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,one_one(A)),one_one(A)))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,groups7121269368397514597t_prod(B,A,H,S)),groups7121269368397514597t_prod(B,A,G,S))) ) ) ) ) ) ).

% prod.related
tff(fact_4901_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S4: set(B),T5: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T6: 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),T5))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S4))
                         => ( aa(B,A,G,A4) = zero_zero(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T5))
                           => ( aa(C,A,H,B3) = zero_zero(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H),T6) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4902_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,A4)),aa(B,A,G,A4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,B4))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_4903_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,F2,B4))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_4904_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T5: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T6: 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),T5))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S4))
                         => ( aa(B,A,G,A4) = one_one(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T5))
                           => ( aa(C,A,H,B3) = one_one(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(C,A,H,T6) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_4905_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
            & ( aa(A,nat,F2,X5) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X5 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4906_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = one_one(nat) )
      <=> ? [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
            & ( aa(A,nat,F2,X5) = one_one(nat) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X5 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4907_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F2: fun(B,A),B4: A,I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S2))
               => 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),S2) = B4 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),B4)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4908_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S2))
               => 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),S2) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S2))
               => ( aa(B,A,F2,I) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4909_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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)),A3),collect(B,aTP_Lamp_ke(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4910_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aTP_Lamp_kf(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_4911_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),collect(B,aTP_Lamp_kg(fun(B,A),fun(B,bool),G)))) = groups7121269368397514597t_prod(B,A,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4912_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N5))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N5))
               => ( 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),N5)) ) ) ) ).

% sums_finite
tff(fact_4913_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_jr(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_4914_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),A3))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_js(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_4915_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite(nat),N5))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N5))
               => ( 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),N5) ) ) ) ) ).

% suminf_finite
tff(fact_4916_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_kh(A,fun(A,bool),A2)))) ) ).

% finite_abs_int_segment
tff(fact_4917_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4918_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4919_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% divmod_abs_code(6)
tff(fact_4920_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_4921_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)),groups7121269368397514597t_prod(A,B,F2,I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4922_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C3: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B4) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4923_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C3: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C3) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4924_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T6) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4925_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4926_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T6) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4927_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4928_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = 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)),A3),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_4929_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A3))
           => ( 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)),A3),B4)) = 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),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B4)) ) ) ) ) ).

% sum_diff
tff(fact_4930_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( groups7121269368397514597t_prod(B,A,G,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4))),groups7121269368397514597t_prod(B,A,G,B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_4931_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G,T6) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4932_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,H,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(B,A,H,T6) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4933_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G,T6) = groups7121269368397514597t_prod(B,A,G,S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4934_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(B,A,G,T6) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4935_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G,C3) = groups7121269368397514597t_prod(B,A,H,C3) )
                   => ( groups7121269368397514597t_prod(B,A,G,A3) = groups7121269368397514597t_prod(B,A,H,B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4936_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G,A3) = groups7121269368397514597t_prod(B,A,H,B4) )
                  <=> ( groups7121269368397514597t_prod(B,A,G,C3) = groups7121269368397514597t_prod(B,A,H,C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4937_sum__diff__nat,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
       => ( 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)),A3),B4)) = 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),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_4938_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_ki(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_4939_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S4: set(B),T5: set(C),H: fun(B,C),S: set(B),T6: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T5))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S4))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = zero_zero(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T5))
                     => ( aa(C,A,G,B3) = 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_kj(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T6) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4940_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A3: set(nat),S4: A,F2: fun(nat,A)] :
          ( sums(A,G,S)
         => ( pp(aa(set(nat),bool,finite_finite(nat),A3))
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S),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_kk(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kl(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4941_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T5: set(C),H: fun(B,C),S: set(B),T6: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T5))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T6),T5))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S4))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = one_one(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T5))
                     => ( aa(C,A,G,B3) = one_one(A) ) )
                 => ( groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_km(fun(B,C),fun(fun(C,A),fun(B,A)),H),G),S) = groups7121269368397514597t_prod(C,A,G,T6) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4942_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4))
           => ( ! [B3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A3)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,B3))) )
             => 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),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B4))) ) ) ) ) ).

% sum_mono2
tff(fact_4943_even__prod__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),groups7121269368397514597t_prod(B,A,F2,A3)))
          <=> ? [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(B,A,F2,X5))) ) ) ) ) ).

% even_prod_iff
tff(fact_4944_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_4945_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_4946_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F2,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3))) )
                 => 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),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B4))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4947_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
           => ( ! [B3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,B3))) )
             => ( ! [A4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A4))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B4))) ) ) ) ) ) ).

% prod_mono2
tff(fact_4948_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),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_kn(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_4949_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( 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,A3)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) ) ) ).

% even_set_encode_iff
tff(fact_4950_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_ko(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4951_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_ko(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_4952_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_kp(nat,fun(nat,bool)),K)))) ).

% finite_Collect_le_nat
tff(fact_4953_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(set(A)),bool,finite_finite(set(A)),collect(set(A),aTP_Lamp_kq(set(A),fun(set(A),bool),A3)))) ) ).

% finite_Collect_subsets
tff(fact_4954_finite__Diff2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( 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)),A3),B4)))
      <=> pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% finite_Diff2
tff(fact_4955_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_kr(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int1
tff(fact_4956_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_ks(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int4
tff(fact_4957_finite__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => 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)),A3),B4))) ) ).

% finite_Diff
tff(fact_4958_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_kt(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int2
tff(fact_4959_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_ku(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int3
tff(fact_4960_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_ep(nat,fun(complex,fun(complex,bool)),N),C2)))) ) ).

% finite_nth_roots
tff(fact_4961_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_4962_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite(int),collect(int,aTP_Lamp_kv(int,fun(int,bool),I)))) ) ).

% finite_divisors_int
tff(fact_4963_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4964_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4965_rev__finite__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% rev_finite_subset
tff(fact_4966_infinite__super,axiom,
    ! [A: $tType,S: set(A),T6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T6))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),S))
       => ~ pp(aa(set(A),bool,finite_finite(A),T6)) ) ) ).

% infinite_super
tff(fact_4967_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( pp(aa(set(A),bool,finite_finite(A),B4))
       => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% finite_subset
tff(fact_4968_Diff__infinite__finite,axiom,
    ! [A: $tType,T6: set(A),S: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),T6))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),S))
       => ~ 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)),S),T6))) ) ) ).

% Diff_infinite_finite
tff(fact_4969_finite__nat__iff__bounded__le,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),set_ord_atMost(nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_4970_finite__nat__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S))
     => ? [K2: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),set_ord_lessThan(nat,K2))) ) ).

% finite_nat_bounded
tff(fact_4971_finite__nat__iff__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),set_ord_lessThan(nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_4972_infinite__int__iff__unbounded__le,axiom,
    ! [S: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite(int),S))
    <=> ! [M5: int] :
        ? [N4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M5),aa(int,int,abs_abs(int),N4)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N4),S)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_4973_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),S))
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N4))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N4),S)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_4974_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_4975_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X7: 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)),X7))
     => ( pp(aa(set(A),bool,finite_finite(A),X7))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X7) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_4976_set__encode__insert,axiom,
    ! [A3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A3))
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,N),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_4977_insert__absorb2,axiom,
    ! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,X),A3)) = aa(set(A),set(A),insert(A,X),A3) ).

% insert_absorb2
tff(fact_4978_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,B2),A3)))
    <=> ( ( A2 = B2 )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ).

% insert_iff
tff(fact_4979_insertCI,axiom,
    ! [A: $tType,A2: A,B4: set(A),B2: A] :
      ( ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
       => ( A2 = B2 ) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,B2),B4))) ) ).

% insertCI
tff(fact_4980_insert__subset,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( 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),A3)),B4))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ) ).

% insert_subset
tff(fact_4981_Diff__insert0,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ) ) ).

% Diff_insert0
tff(fact_4982_insert__Diff1,axiom,
    ! [A: $tType,X: A,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A3)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ) ) ).

% insert_Diff1
tff(fact_4983_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: 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)),A3),aa(set(A),set(A),insert(A,A2),B4))))
    <=> 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)),A3),B4))) ) ).

% finite_Diff_insert
tff(fact_4984_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_4985_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),insert(B,X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_4986_insert__Diff__if,axiom,
    ! [A: $tType,X: A,B4: set(A),A3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A3)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A3)),B4) = 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)),A3),B4)) ) ) ) ).

% insert_Diff_if
tff(fact_4987_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),B4))) ) ).

% subset_insertI2
tff(fact_4988_subset__insertI,axiom,
    ! [A: $tType,B4: set(A),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(A),set(A),insert(A,A2),B4))) ).

% subset_insertI
tff(fact_4989_subset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,X),B4)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ) ).

% subset_insert
tff(fact_4990_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D4: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),D4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A2),C3)),aa(set(A),set(A),insert(A,A2),D4))) ) ).

% insert_mono
tff(fact_4991_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ? [B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,A2),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_4992_insert__commute,axiom,
    ! [A: $tType,X: A,Y4: A,A3: set(A)] : aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y4),A3)) = aa(set(A),set(A),insert(A,Y4),aa(set(A),set(A),insert(A,X),A3)) ).

% insert_commute
tff(fact_4993_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A,B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B4))
       => ( ( aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),insert(A,B2),B4) )
        <=> ( ( ( A2 = B2 )
             => ( A3 = B4 ) )
            & ( ( A2 != B2 )
             => ? [C7: set(A)] :
                  ( ( A3 = aa(set(A),set(A),insert(A,B2),C7) )
                  & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),C7))
                  & ( B4 = aa(set(A),set(A),insert(A,A2),C7) )
                  & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C7)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4994_insert__absorb,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),insert(A,A2),A3) = A3 ) ) ).

% insert_absorb
tff(fact_4995_insert__ident,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
       => ( ( aa(set(A),set(A),insert(A,X),A3) = aa(set(A),set(A),insert(A,X),B4) )
        <=> ( A3 = B4 ) ) ) ) ).

% insert_ident
tff(fact_4996_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ~ ! [B8: set(A)] :
            ( ( A3 = aa(set(A),set(A),insert(A,X),B8) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B8)) ) ) ).

% Set.set_insert
tff(fact_4997_insertI2,axiom,
    ! [A: $tType,A2: A,B4: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,B2),B4))) ) ).

% insertI2
tff(fact_4998_insertI1,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,A2),B4))) ).

% insertI1
tff(fact_4999_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,B2),A3)))
     => ( ( A2 != B2 )
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ).

% insertE
tff(fact_5000_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] : aa(set(A),set(A),insert(A,A2),collect(A,P)) = collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_kw(A,fun(fun(A,bool),fun(A,bool)),A2),P)) ).

% insert_Collect
tff(fact_5001_insert__compr,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_kx(A,fun(set(A),fun(A,bool)),A2),B4)) ).

% insert_compr
tff(fact_5002_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),X: A,C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),insert(A,X),C3))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C3)))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_5003_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_5004_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_5005_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),insert(B,X),A3)) = groups7121269368397514597t_prod(B,A,G,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),insert(B,X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,A3)) ) ) ) ) ) ).

% prod.insert_if
tff(fact_5006_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_5007_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_5008_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5009_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5010_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5011_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_5012_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_5013_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_5014_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_5015_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_5016_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_5017_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_5018_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_5019_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_ky(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_kz(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_5020_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X7: set(A),F2: fun(A,nat)] :
      ( 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)),X7))
     => ( pp(aa(set(A),bool,finite_finite(A),X7))
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs)) = 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_la(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X7) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_5021_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)))
          <=> 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(B),nat,finite_card(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_lb(set(B),fun(fun(B,A),fun(B,bool)),A3),F2))))) ) ) ) ).

% even_sum_iff
tff(fact_5022_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_5023_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_5024_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_kp(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_5025_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_5026_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_5027_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) )
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Ns)))
             => ( X5 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_5028_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_5029_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_lc(B,A),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_5030_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_5031_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y4: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ld(A,fun(B,A),Y4)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),Y4) ) ).

% sum_constant
tff(fact_5032_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_5033_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_5034_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),collect(set(A),aa(nat,fun(set(A),bool),aTP_Lamp_le(set(A),fun(nat,fun(set(A),bool)),A3),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A3)),K) ) ) ).

% n_subsets
tff(fact_5035_card__subset__eq,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),B4) )
         => ( A3 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_5036_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [B8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = N )
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_5037_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A3: set(B),R2: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( ! [A4: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
           => ? [B9: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B9),B4))
                & pp(aa(A,bool,aa(B,fun(A,bool),R2,A4),B9)) ) )
       => ( ! [A12: B,A23: B,B3: A] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A12),A3))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A23),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B4))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R2,A12),B3))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R2,A23),B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),aa(set(A),nat,finite_card(A),B4))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_5038_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A3)))) ).

% card_insert_le
tff(fact_5039_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [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(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% member_le_sum_list
tff(fact_5040_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ju(set(A),fun(nat,fun(list(A),bool)),A3),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_5041_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),C2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_es(fun(B,A),fun(A,fun(B,A)),F2),C2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_5042_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_et(A,fun(fun(B,A),fun(B,A)),C2),F2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F2,Xs))) ) ).

% sum_list_const_mult
tff(fact_5043_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_eu(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_5044_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_lf(nat,fun(nat,fun(list(nat),bool)),M),N5))) = 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_lg(nat,fun(nat,fun(list(nat),bool)),M),N5)))),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_lh(nat,fun(nat,fun(list(nat),bool)),M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5045_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_lf(nat,fun(nat,fun(list(nat),bool)),M),N5))) = 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),N5),M)),one_one(nat))),N5) ).

% card_length_sum_list
tff(fact_5046_card__2__iff_H,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
          & ? [Xa4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S))
              & ( X5 != Xa4 )
              & ! [Xb3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xb3),S))
                 => ( ( Xb3 = X5 )
                    | ( Xb3 = Xa4 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5047_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
     => pp(aa(set(A),bool,finite_finite(A),A3)) ) ).

% card_ge_0_finite
tff(fact_5048_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A3)) = aa(set(A),nat,finite_card(A),A3) ) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ) ).

% card_insert_if
tff(fact_5049_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B7),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & pp(aa(set(A),bool,finite_finite(A),B10)) ) ) ).

% card_Suc_eq_finite
tff(fact_5050_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N: nat,S: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S)))
     => ~ ! [T7: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T7),S))
           => ( ( aa(set(A),nat,finite_card(A),T7) = N )
             => ~ pp(aa(set(A),bool,finite_finite(A),T7)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_5051_card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))) ) ) ).

% card_mono
tff(fact_5052_card__seteq,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B4)),aa(set(A),nat,finite_card(A),A3)))
         => ( A3 = B4 ) ) ) ) ).

% card_seteq
tff(fact_5053_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F3: set(A),C3: nat] :
      ( ! [G2: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G2),F3))
         => ( pp(aa(set(A),bool,finite_finite(A),G2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G2)),C3)) ) )
     => ( pp(aa(set(A),bool,finite_finite(A),F3))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F3)),C3)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_5054_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)))
         => 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)),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_5055_card__le__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)))
         => 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)),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_5056_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_5057_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M10: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M10))
     => ? [H3: fun(A,nat)] : bij_betw(A,nat,H3,M10,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M10))) ) ).

% ex_bij_betw_finite_nat
tff(fact_5058_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),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_5059_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
               => ( X5 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_5060_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => 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_5061_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs)))) ) ).

% sum_list_abs
tff(fact_5062_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_5063_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_5064_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_li(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_lj(set(nat),fun(nat,fun(nat,bool)),M10),I))) ) ) ).

% card_less_Suc2
tff(fact_5065_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_li(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_lj(set(nat),fun(nat,fun(nat,bool)),M10),I))) ) ) ).

% card_less_Suc
tff(fact_5066_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_lj(set(nat),fun(nat,fun(nat,bool)),M10),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_5067_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_5068_sum__constant__scaleR,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y4: A,A3: set(C)] : aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aTP_Lamp_lk(A,fun(C,A),Y4)),A3) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,semiring_1_of_nat(real),aa(set(C),nat,finite_card(C),A3))),Y4) ) ).

% sum_constant_scaleR
tff(fact_5069_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ll(fun(A,nat),fun(A,nat),F2)),A3) = 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),A3)),aa(set(A),nat,finite_card(A),A3)) ).

% sum_Suc
tff(fact_5070_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T6: set(B),R: fun(A,fun(B,bool)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),S))
     => ( pp(aa(set(B),bool,finite_finite(B),T6))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),T6))
             => ( aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_lm(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S),R),X3))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_lo(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T6),R)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T6)) ) ) ) ) ).

% sum_multicount
tff(fact_5071_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3)))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_5072_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs)))) ) ) ).

% sum_list_mono
tff(fact_5073_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A3)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_lp(A,real)),A3) ).

% real_of_card
tff(fact_5074_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F2: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),K5)) )
         => 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),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),K5))) ) ) ).

% sum_bounded_above
tff(fact_5075_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),K5: A,F2: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),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),A3))),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))) ) ) ).

% sum_bounded_below
tff(fact_5076_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
           => ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( X5 = Xa4 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_5077_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A3: 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),A3)))
    <=> ? [A6: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,A6),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),B10))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B10)))
          & pp(aa(set(A),bool,finite_finite(A),B10)) ) ) ).

% card_le_Suc_iff
tff(fact_5078_card__Diff__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_Diff_subset
tff(fact_5079_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4)) ) ) ) ).

% card_psubset
tff(fact_5080_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => 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),A3)),aa(set(A),nat,finite_card(A),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)))) ) ).

% diff_card_le_card_Diff
tff(fact_5081_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_jw(set(A),fun(nat,fun(list(A),bool)),A3),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),A3))),set_ord_atMost(nat,N)) ) ) ).

% card_lists_length_le
tff(fact_5082_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M10: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M10))
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M10)),M10) ) ).

% ex_bij_betw_nat_finite
tff(fact_5083_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_5084_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_ki(nat,fun(A,bool),N)))),N)) ) ) ).

% card_roots_unity
tff(fact_5085_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = 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,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5086_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),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),N5)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_5087_card__sum__le__nat__sum,axiom,
    ! [S: 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_gh(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(nat,nat)),S))) ).

% card_sum_le_nat_sum
tff(fact_5088_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_ll(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_5089_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R2: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),map(C,B,aTP_Lamp_lq(B,fun(C,B),R2),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))),R2) ) ).

% sum_list_triv
tff(fact_5090_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_aw(complex,fun(nat,fun(complex,bool)),C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_5091_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_av(nat,fun(complex,bool),N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_5092_sum__norm__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S: set(B),F2: fun(B,A),K5: real] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(B,A,F2,X3))),K5)) )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),S))),K5))) ) ) ).

% sum_norm_bound
tff(fact_5093_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: 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),A3))
             => ( 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),A3)),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),groups7121269368397514597t_prod(B,A,F2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).

% prod_le_power
tff(fact_5094_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F2: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I3)),K5)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A3)))
           => 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),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),K5))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_5095_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_5096_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y4: 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),Y4)),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),Y4))),N)) ) ) ).

% card_insert_le_m1
tff(fact_5097_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_lr(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_5098_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_ko(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ).

% polyfun_roots_card
tff(fact_5099_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A2: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ls(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),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),S)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ls(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2),S) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_5100_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_ko(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_ko(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ) ).

% polyfun_rootbound
tff(fact_5101_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_5102_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y4 )
     => ( ( ( 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)))))) )
         => ( Y4 = 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)))))) )
         => ( Y4 = 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_5103_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_lt(set(A),fun(nat,fun(list(A),bool)),A3),K))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_gh(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_5104_bot__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( bot(C)
     => ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).

% bot_apply
tff(fact_5105_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( bot_bot(set(A)) = collect(A,P) )
    <=> ! [X5: A] : ~ pp(aa(A,bool,P,X5)) ) ).

% empty_Collect_eq
tff(fact_5106_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ! [X5: A] : ~ pp(aa(A,bool,P,X5)) ) ).

% Collect_empty_eq
tff(fact_5107_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X5: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_5108_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),bot_bot(set(A)))) ).

% empty_iff
tff(fact_5109_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A3)) ).

% empty_subsetI
tff(fact_5110_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A))))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_5111_singletonI,axiom,
    ! [A: $tType,A2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ).

% singletonI
tff(fact_5112_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_5113_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_5114_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_5115_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_5116_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_5117_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : collect(A,aa(A,fun(A,bool),fequal(A),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_5118_singleton__conv,axiom,
    ! [A: $tType,A2: A] : collect(A,aTP_Lamp_lu(A,fun(A,bool),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_5119_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_5120_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_5121_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_5122_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) = aa(set(A),set(A),insert(A,A2),A3) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5123_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5124_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_5125_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_5126_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_5127_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A2),A3) ).

% insert_Diff_single
tff(fact_5128_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_5129_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_5130_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),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),A3)) ) ).

% subset_Compl_singleton
tff(fact_5131_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_5132_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_lt(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_5133_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) = aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_5134_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_5135_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,C2),aa(set(A),set(A),insert(A,D2),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_5136_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_5137_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_5138_Collect__conv__if2,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lv(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lv(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_5139_Collect__conv__if,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lw(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lw(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_5140_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),bot_bot(set(A)))) ).

% not_psubset_empty
tff(fact_5141_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A2)) ) ) ).

% bot.not_eq_extremum
tff(fact_5142_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_5143_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A2)) ) ).

% bot.extremum
tff(fact_5144_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_5145_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_5146_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X4: A] : aa(A,B,bot_bot(fun(A,B)),X4) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_5147_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X5: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_5148_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_5149_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ).

% equals0D
tff(fact_5150_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),bot_bot(set(A)))) ).

% emptyE
tff(fact_5151_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_lx(A,bool)) ).

% empty_def
tff(fact_5152_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ).

% diff_shunt_var
tff(fact_5153_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5154_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5155_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_5156_subset__singleton__iff,axiom,
    ! [A: $tType,X7: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))
    <=> ( ( X7 = bot_bot(set(A)) )
        | ( X7 = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_5157_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,X),A3)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_5158_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))),B4) ).

% Diff_insert2
tff(fact_5159_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_5160_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_5161_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_5162_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S: set(B),P: fun(set(B),bool),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( pp(aa(set(B),bool,P,bot_bot(set(B))))
           => ( ! [X3: B,S5: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite(B),S5))
                 => ( ! [Y3: B] :
                        ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),S5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,Y3)),aa(B,A,F2,X3))) )
                   => ( pp(aa(set(B),bool,P,S5))
                     => pp(aa(set(B),bool,P,aa(set(B),set(B),insert(B,X3),S5))) ) ) )
             => pp(aa(set(B),bool,P,S)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_5163_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),F3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F3),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F4: set(A)] :
                ( pp(aa(set(A),bool,finite_finite(A),F4))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A3))
                   => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F4))
                     => ( pp(aa(set(A),bool,P,F4))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,A4),F4))) ) ) ) ) )
           => pp(aa(set(A),bool,P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5164_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),F3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F3),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F4: set(A)] :
                ( pp(aa(set(A),bool,finite_finite(A),F4))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F4))
                   => ( pp(aa(set(A),bool,P,F4))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,A4),F4))) ) ) ) )
           => pp(aa(set(A),bool,P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_5165_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% card_eq_0_iff
tff(fact_5166_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,P,A3))
       => ( ! [A4: A,A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A7))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A7))
               => ( pp(aa(set(A),bool,P,A7))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,A4),bot_bot(set(A)))))) ) ) )
         => pp(aa(set(A),bool,P,bot_bot(set(A)))) ) ) ) ).

% finite_empty_induct
tff(fact_5167_infinite__coinduct,axiom,
    ! [A: $tType,X7: fun(set(A),bool),A3: set(A)] :
      ( pp(aa(set(A),bool,X7,A3))
     => ( ! [A7: set(A)] :
            ( pp(aa(set(A),bool,X7,A7))
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A7))
                & ( pp(aa(set(A),bool,X7,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))))
                  | ~ 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)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A)))))) ) ) )
       => ~ pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% infinite_coinduct
tff(fact_5168_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A2: A] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S))
     => ~ 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)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ).

% infinite_remove
tff(fact_5169_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,X),B4))) ) ).

% Diff_single_insert
tff(fact_5170_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,X),B4)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B4)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_5171_Compl__insert,axiom,
    ! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_5172_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_5173_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
    <=> ( ( A3 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% card_gt_0_iff
tff(fact_5174_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B7),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_5175_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B3: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B3),B8) )
          & ~ 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_eq_SucD
tff(fact_5176_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X5: A] : A3 = aa(set(A),set(A),insert(A,X5),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_5177_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B4: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite(A),B4))
         => pp(aa(set(A),bool,P,B4)) )
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),B4))
                 => ( ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% remove_induct
tff(fact_5178_finite__remove__induct,axiom,
    ! [A: $tType,B4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),B4))
                 => ( ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_5179_card__Diff1__le,axiom,
    ! [A: $tType,A3: 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ).

% card_Diff1_le
tff(fact_5180_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),S))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [T7: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T7),S))
             => ( pp(aa(set(A),bool,P,T7))
               => ? [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T7)))
                    & pp(aa(set(A),bool,P,aa(set(A),set(A),insert(A,X4),T7))) ) ) )
         => pp(aa(set(A),bool,P,S)) ) ) ) ).

% finite_induct_select
tff(fact_5181_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),aa(set(A),set(A),insert(A,X),B4)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B4)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B4)) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5182_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5183_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_5184_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_5185_sum__diff1__nat,axiom,
    ! [A: $tType,A2: A,A3: set(A),F2: fun(A,nat)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( 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)),A3),aa(set(A),set(A),insert(A,A2),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),A3)),aa(A,nat,F2,A2)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( 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)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) ) ) ) ).

% sum_diff1_nat
tff(fact_5186_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
               => ( 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))) ) )
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ) ) ).

% prod_mono_strict
tff(fact_5187_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X5: A,Y5: A] :
          ( ( S = aa(set(A),set(A),insert(A,X5),aa(set(A),set(A),insert(A,Y5),bot_bot(set(A)))) )
          & ( X5 != Y5 ) ) ) ).

% card_2_iff
tff(fact_5188_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,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)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_5189_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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),A3)) = 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)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_5190_sum__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),A2: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
             => ( 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)),A3),aa(set(B),set(B),insert(B,A2),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),A3)),aa(B,A,F2,A2)) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
             => ( 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)),A3),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3) ) ) ) ) ) ).

% sum_diff1
tff(fact_5191_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X5: A,Y5: A,Z5: A] :
          ( ( S = aa(set(A),set(A),insert(A,X5),aa(set(A),set(A),insert(A,Y5),aa(set(A),set(A),insert(A,Z5),bot_bot(set(A))))) )
          & ( X5 != Y5 )
          & ( Y5 != Z5 )
          & ( X5 != Z5 ) ) ) ).

% card_3_iff
tff(fact_5192_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(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(set(A),nat,finite_card(A),A3)))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_5193_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),insert(B,X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_5194_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( groups7121269368397514597t_prod(B,A,G,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_5195_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_5196_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_5197_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_5198_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_Diff1_less
tff(fact_5199_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),X: A,Y4: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
         => 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Y4),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_Diff2_less
tff(fact_5200_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)))
    <=> ( pp(aa(set(A),bool,finite_finite(A),A3))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_5201_card__Diff__singleton__if,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_5202_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_5203_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( 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_ly(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A2)),aa(set(B),A,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)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,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_ly(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S) = 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)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_5204_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lz(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),groups7121269368397514597t_prod(B,A,C2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lz(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2),S) = groups7121269368397514597t_prod(B,A,C2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_5205_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: C,A3: set(C),F2: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I),A3))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A3),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,X3))) )
           => ( pp(aa(set(C),bool,finite_finite(C),A3))
             => 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),A3))) ) ) ) ) ).

% member_le_sum
tff(fact_5206_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A3: set(B),F2: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => 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),K5),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))))) )
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( ( A3 != 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),A3)),K5)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5207_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A3: set(B),F2: fun(B,A),A2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( aa(B,A,F2,A2) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),aa(B,A,F2,A2)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) = groups7121269368397514597t_prod(B,A,F2,A3) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_5208_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,exp(A),X)),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_5209_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(set(A),fun(list(A),bool),aTP_Lamp_ma(nat,fun(set(A),fun(list(A),bool)),K),A3))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_gh(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_5210_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa(product_prod(int,int),bool,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_5211_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y4 )
     => ( pp(aa(product_prod(int,int),bool,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)))))) )
               => ( Y4 = 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)))))) )
               => ( Y4 = 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)))))) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,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_5212_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,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] :
            ( pp(aa(product_prod(int,int),bool,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_5213_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_5214_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_5215_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_5216_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_5217_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),insert(nat,M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_5218_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_5219_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,bot_bot(fun(A,bool))) ).

% bot_set_def
tff(fact_5220_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_5221_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_5222_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_5223_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_5224_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,M,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_5225_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_5226_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_5227_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_5228_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_5229_the__elem__def,axiom,
    ! [A: $tType,X7: set(A)] : the_elem(A,X7) = the(A,aTP_Lamp_mb(set(A),fun(A,bool),X7)) ).

% the_elem_def
tff(fact_5230_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_5231_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_13: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_13))
     => ( ! [X3: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X3))
           => ? [Y3: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y3))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X3),Y3)) ) )
       => ? [F5: fun(nat,A)] :
          ! [N7: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N7),aa(nat,A,F5,N7)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N7),aa(nat,A,F5,N7)),aa(nat,A,F5,aa(nat,nat,suc,N7)))) ) ) ) ).

% dependent_nat_choice
tff(fact_5232_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite(A),F3))
         => ( 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_mc(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))),F3))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),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),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),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))))) = groups1027152243600224163dd_sum(A,B,F2,I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_5233_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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,A2),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_5234_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_5235_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_5236_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_5237_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) ).

% length_upt
tff(fact_5238_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),insert(B,I),I5)) = groups1027152243600224163dd_sum(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),insert(B,I),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P3,I)),groups1027152243600224163dd_sum(B,A,P3,I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_5239_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_5240_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map(nat,nat,suc,upt(M,N)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_5241_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,bool),aTP_Lamp_md(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_5242_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),X: A,I: nat] :
      ( 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)),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_5243_map__add__upt,axiom,
    ! [N: nat,M: nat] : map(nat,nat,aTP_Lamp_me(nat,fun(nat,nat),N),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_5244_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : map(nat,A,aTP_Lamp_mf(A,fun(nat,A),X),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_5245_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: 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),list_update(A,Xs,I,X))),aa(set(A),set(A),insert(A,X),aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_5246_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map(nat,nat,aTP_Lamp_mg(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_5247_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1337092689740270186AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_5248_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_5249_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_5250_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T6: 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)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,T6) = groups1027152243600224163dd_sum(B,A,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_5251_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T6: 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)),S),T6))
         => ( ! [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)),T6),S)))
               => ( aa(B,A,H,I3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,S) = groups1027152243600224163dd_sum(B,A,H,T6) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_5252_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,T6) = groups1027152243600224163dd_sum(B,A,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_5253_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,S) = groups1027152243600224163dd_sum(B,A,G,T6) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_5254_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_jx(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_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mh(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_5255_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jx(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_5256_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M: 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),M)))
     => ( aa(nat,A,nth(A,map(nat,A,F2,upt(M,N))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_5257_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_5258_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) )
     => ( ! [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),M),I3)) ) )
       => ( map(nat,A,F2,upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_5259_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_5260_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs),I)),bot_bot(set(A))))))
       => distinct(A,list_update(A,Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_5261_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_mc(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),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),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)) ) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),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))))) = groups1027152243600224163dd_sum(A,B,F2,I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_5262_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F2: fun(nat,B),Ns: list(nat)] :
          ( ! [X3: nat,Y: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F2,X3)),aa(nat,B,F2,Y))) )
         => ( 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_5263_is__singleton__the__elem,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( A3 = aa(set(A),set(A),insert(A,the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_5264_card__Pow,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Pow
tff(fact_5265_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_5266_Pow__singleton__iff,axiom,
    ! [A: $tType,X7: set(A),Y6: set(A)] :
      ( ( pow2(A,X7) = aa(set(set(A)),set(set(A)),insert(set(A),Y6),bot_bot(set(set(A)))) )
    <=> ( ( X7 = bot_bot(set(A)) )
        & ( Y6 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_5267_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% Pow_iff
tff(fact_5268_PowI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B4))) ) ).

% PowI
tff(fact_5269_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_5270_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N,X)) ) ).

% sorted_replicate
tff(fact_5271_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) = collect(set(A),aTP_Lamp_kq(set(A),fun(set(A),bool),A3)) ).

% Pow_def
tff(fact_5272_Pow__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow2(A,A3)),pow2(A,B4))) ) ).

% Pow_mono
tff(fact_5273_PowD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B4)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% PowD
tff(fact_5274_Pow__top,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,A3))) ).

% Pow_top
tff(fact_5275_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_5276_Pow__bottom,axiom,
    ! [A: $tType,B4: set(A)] : pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),bot_bot(set(A))),pow2(A,B4))) ).

% Pow_bottom
tff(fact_5277_Pow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_5278_sorted__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N)) ).

% sorted_upt
tff(fact_5279_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
        <=> sorted_wrt(B,aTP_Lamp_mi(fun(B,A),fun(B,fun(B,bool)),F2),Xs) ) ) ).

% sorted_map
tff(fact_5280_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X3: A,Y: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
             => ( X3 = Y ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_5281_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_5282_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5283_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_5284_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_5285_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_5286_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & distinct(A,Y3) )
                 => ( Y3 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5287_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_5288_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_5289_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),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),J))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_5290_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_5291_is__singleton__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ? [X5: A] : A3 = aa(set(A),set(A),insert(A,X5),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_5292_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X3: A] : A3 != aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_5293_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_mj(nat,fun(nat,fun(set(nat),bool)),N),K))) ).

% binomial_def
tff(fact_5294_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)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,map(B,A,F2,L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_5295_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_5296_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_5297_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_5298_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_5299_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ).

% fst_divmod_nat
tff(fact_5300_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M,N)) = modulo_modulo(nat,M,N) ).

% snd_divmod_nat
tff(fact_5301_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_5302_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_5303_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_5304_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) ) ).

% fst_divmod
tff(fact_5305_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,N)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)) ) ).

% snd_divmod
tff(fact_5306_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: 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)),P3),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),P3)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),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),P3)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_5307_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)),S: set(B),F2: fun(B,A),A3: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( sorted_wrt(A,Less,map(B,A,F2,L))
              & ( aa(list(B),set(B),set2(B),L) = A3 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A3) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A3) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_5308_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S))
       => ( sorted_wrt(A,Less_eq,map(B,A,F2,Xs))
         => ( distinct(B,Xs)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_5309_quotient__of__denom__pos_H,axiom,
    ! [R2: 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(R2)))) ).

% quotient_of_denom_pos'
tff(fact_5310_bezw__non__0,axiom,
    ! [Y4: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y4))
     => ( bezw(X,Y4) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y4))))) ) ) ).

% bezw_non_0
tff(fact_5311_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B),B4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),S))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A3) = sorted8670434370408473282of_set(A,B,Less_eq,F2,B4) )
           => ( pp(aa(set(B),bool,finite_finite(B),A3))
             => ( pp(aa(set(B),bool,finite_finite(B),B4))
               => ( A3 = B4 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_5312_bezw_Osimps,axiom,
    ! [Y4: nat,X: nat] :
      ( ( ( Y4 = zero_zero(nat) )
       => ( bezw(X,Y4) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
      & ( ( Y4 != zero_zero(nat) )
       => ( bezw(X,Y4) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y4,modulo_modulo(nat,X,Y4)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y4))))) ) ) ) ).

% bezw.simps
tff(fact_5313_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y4: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y4 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y4 = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y4 = 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_5314_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs))) ).

% sorted_enumerate
tff(fact_5315_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) = A3 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_5316_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)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_5317_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => distinct(A,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_5318_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => sorted_wrt(A,Less,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_5319_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => sorted_wrt(A,Less_eq,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_5320_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y4: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y4 )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y4 = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y4 = 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))))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))) ) ) ) ).

% bezw.pelims
tff(fact_5321_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_5322_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_5323_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),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,M,N))) ).

% minus_numeral_mod_numeral
tff(fact_5324_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_5325_Divides_Oadjust__mod__def,axiom,
    ! [R2: int,L: int] :
      ( ( ( R2 = zero_zero(int) )
       => ( adjust_mod(L,R2) = zero_zero(int) ) )
      & ( ( R2 != zero_zero(int) )
       => ( adjust_mod(L,R2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R2) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_5326_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)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,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),A3)),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),insert(B,X),A3)) = 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)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_5327_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,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),A3)),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),insert(B,X),A3)) = insort_key(A,B,Less_eq,F2,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_5328_normalize__def,axiom,
    ! [P3: 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),P3)))
       => ( normalize(P3) = 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),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))) ) )
      & ( ~ 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),P3)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int) )
           => ( normalize(P3) = 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),P3) != zero_zero(int) )
           => ( normalize(P3) = 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),P3)),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),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),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),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_5329_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_5330_gcd__dvd1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: 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),A2),B2)),A2)) ) ).

% gcd_dvd1
tff(fact_5331_gcd__dvd2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: 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),A2),B2)),B2)) ) ).

% gcd_dvd2
tff(fact_5332_gcd__greatest__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2)) ) ) ) ).

% gcd_greatest_iff
tff(fact_5333_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A2) ) ).

% gcd_neg_numeral_1
tff(fact_5334_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_5335_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: 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),A2),B2)),one_one(A)))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_5336_gcd__pos__int,axiom,
    ! [M: 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),M),N)))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_5337_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_5338_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_5339_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_5340_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_5341_gcd__proj2__if__dvd__int,axiom,
    ! [Y4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Y4),X))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4) = aa(int,int,abs_abs(int),Y4) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_5342_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),Y4))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4) = aa(int,int,abs_abs(int),X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_5343_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add_mult
tff(fact_5344_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))) ) ).

% gcd_dvd_prod
tff(fact_5345_gcd__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D2))) ) ) ) ).

% gcd_mono
tff(fact_5346_dvd__gcdD1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% dvd_gcdD1
tff(fact_5347_dvd__gcdD2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2)) ) ) ).

% dvd_gcdD2
tff(fact_5348_gcd__dvdI1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),C2)) ) ) ).

% gcd_dvdI1
tff(fact_5349_gcd__dvdI2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( 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),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),C2)) ) ) ).

% gcd_dvdI2
tff(fact_5350_gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( 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),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ) ).

% gcd_greatest
tff(fact_5351_gcd__dvd__antisym,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)))
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D2) ) ) ) ) ).

% gcd_dvd_antisym
tff(fact_5352_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M: 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),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_diff1
tff(fact_5353_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: A,M: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),M)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_diff2
tff(fact_5354_gcd__ge__0__int,axiom,
    ! [X: int,Y4: 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),Y4))) ).

% gcd_ge_0_int
tff(fact_5355_bezout__int,axiom,
    ! [X: int,Y4: int] :
    ? [U2: int,V4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y4)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4) ).

% bezout_int
tff(fact_5356_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)) ).

% gcd_mult_distrib_int
tff(fact_5357_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_5358_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_5359_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_5360_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_5361_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2)) ) ).

% gcd_le1_int
tff(fact_5362_gcd__le2__int,axiom,
    ! [B2: int,A2: 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),A2),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_5363_gcd__cases__int,axiom,
    ! [X: int,Y4: 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)),Y4))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4))) ) )
     => ( ( 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),Y4),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),Y4)))) ) )
       => ( ( 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)),Y4))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y4))) ) )
         => ( ( 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),Y4),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),Y4)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4))) ) ) ) ) ).

% gcd_cases_int
tff(fact_5364_gcd__unique__int,axiom,
    ! [D2: int,A2: 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),A2))
        & 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),A2))
              & 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),A2),B2) ) ) ).

% gcd_unique_int
tff(fact_5365_gcd__non__0__int,axiom,
    ! [Y4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y4))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y4),modulo_modulo(int,X,Y4)) ) ) ).

% gcd_non_0_int
tff(fact_5366_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_5367_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)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,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),A3)),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_5368_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F2,G,P3) = 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),P3))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_5369_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite(set(A)),A3))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
           => pp(aa(set(A),bool,finite_finite(A),X3)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_mk(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_ml(set(set(A)),fun(set(set(A)),bool),A3)))) ) ) ) ).

% card_UNION
tff(fact_5370_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_5371_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_5372_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),zero_zero(nat)) = A2 ).

% gcd_nat.right_neutral
tff(fact_5373_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_5374_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_5375_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_5376_gcd__nat_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb1
tff(fact_5377_gcd__nat_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb2
tff(fact_5378_gcd__nat_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2)) ) ) ).

% gcd_nat.bounded_iff
tff(fact_5379_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),Y4))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_5380_gcd__proj2__if__dvd__nat,axiom,
    ! [Y4: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Y4),X))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4) = Y4 ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_5381_gcd__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_5382_gcd__pos__nat,axiom,
    ! [M: 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),M),N)))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_5383_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),M)),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),M),N)) ).

% gcd_int_int_eq
tff(fact_5384_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y4)) = Y4 ) ) ) ).

% Sup_atLeastAtMost
tff(fact_5385_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y4)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_5386_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_5387_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_5388_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_5389_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% gcd_mult_distrib_nat
tff(fact_5390_gcd__nat_Omono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),D2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),C2),D2))) ) ) ).

% gcd_nat.mono
tff(fact_5391_gcd__nat_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.orderE
tff(fact_5392_gcd__nat_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2)) ) ).

% gcd_nat.orderI
tff(fact_5393_gcd__nat_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb3
tff(fact_5394_gcd__nat_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2))
        & ( B2 != A2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb4
tff(fact_5395_gcd__nat_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2)) ) ) ).

% gcd_nat.boundedE
tff(fact_5396_gcd__nat_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))) ) ) ).

% gcd_nat.boundedI
tff(fact_5397_gcd__nat_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
    <=> ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.order_iff
tff(fact_5398_gcd__nat_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2)) ).

% gcd_nat.cobounded1
tff(fact_5399_gcd__nat_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2)) ).

% gcd_nat.cobounded2
tff(fact_5400_gcd__nat_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb_iff1
tff(fact_5401_gcd__nat_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A2))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb_iff2
tff(fact_5402_gcd__nat_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)) ) ).

% gcd_nat.coboundedI1
tff(fact_5403_gcd__nat_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)) ) ).

% gcd_nat.coboundedI2
tff(fact_5404_gcd__nat_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
        & ( A2 != aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2) ) )
     => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
            & ( A2 != B2 ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
              & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_5405_gcd__nat_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B2))
        & ( A2 != B2 ) )
    <=> ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_5406_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),C2))
        & ( A2 != C2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_5407_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
        & ( B2 != C2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_5408_gcd__unique__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
        & ! [E3: nat] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E3),A2))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E3),B2)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E3),D2)) ) )
    <=> ( D2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_unique_nat
tff(fact_5409_gcd__le2__nat,axiom,
    ! [B2: nat,A2: 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),A2),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_5410_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != 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),A2),B2)),A2)) ) ).

% gcd_le1_nat
tff(fact_5411_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff1_nat
tff(fact_5412_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff2_nat
tff(fact_5413_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y4: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y4 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y4 = X ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_5414_gcd__nat_Osimps,axiom,
    ! [Y4: nat,X: nat] :
      ( ( ( Y4 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4) = X ) )
      & ( ( Y4 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y4),modulo_modulo(nat,X,Y4)) ) ) ) ).

% gcd_nat.simps
tff(fact_5415_gcd__non__0__nat,axiom,
    ! [Y4: nat,X: nat] :
      ( ( Y4 != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y4),modulo_modulo(nat,X,Y4)) ) ) ).

% gcd_non_0_nat
tff(fact_5416_card__Union__le__sum__card,axiom,
    ! [A: $tType,U3: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3))) ).

% card_Union_le_sum_card
tff(fact_5417_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),K),L) = aa(code_integer,code_integer,abs_abs(code_integer),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),L),zero_zero(code_integer)),K,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(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))))) ).

% gcd_code_integer
tff(fact_5418_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs: list(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),remove1(A,X,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_remove1_subset
tff(fact_5419_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_5420_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2)) )
           => 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),S))),A2)) ) ) ) ).

% cInf_abs_ge
tff(fact_5421_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X3: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_5422_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X3: nat,Y: 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),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),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),A2),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_5423_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U3: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),U3))
         => pp(aa(set(A),bool,finite_finite(A),X3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3))) ) ).

% card_Union_le_sum_card_weak
tff(fact_5424_less__by__empty,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B4: set(product_prod(A,A))] :
      ( ( A3 = bot_bot(set(product_prod(A,A))) )
     => 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))),A3),B4)) ) ).

% less_by_empty
tff(fact_5425_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_5426_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => 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),X3),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),S)),L))),E2)) ) ) ) ).

% cInf_asclose
tff(fact_5427_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => 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),X3),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),S)),L))),E2)) ) ) ) ).

% cSup_asclose
tff(fact_5428_gcd__int__def,axiom,
    ! [X: int,Y4: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y4) = 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),Y4)))) ).

% gcd_int_def
tff(fact_5429_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B11: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)))
       => ~ ! [F6: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F6))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F6),B11))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6))) ) ) ) ) ).

% finite_subset_Union
tff(fact_5430_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_5431_bezw__aux,axiom,
    ! [X: nat,Y4: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y4)) = 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,Y4))),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,Y4))),aa(nat,int,semiring_1_of_nat(int),Y4))) ).

% bezw_aux
tff(fact_5432_nat__descend__induct,axiom,
    ! [N: nat,P: fun(nat,bool),M: 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,M)) ) ) ).

% nat_descend_induct
tff(fact_5433_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y4,X)) = Y4 ) ) ) ).

% cInf_atLeastAtMost
tff(fact_5434_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y4,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_5435_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y4: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y4 )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y4 = X ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y4 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,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_5436_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_5437_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,P,A2))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C4),B2))
                  & ! [X4: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),C4)) )
                     => pp(aa(A,bool,P,X4)) )
                  & ! [D6: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),D6)) )
                         => pp(aa(A,bool,P,X3)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D6),C4)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_5438_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X7: set(A),A2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
         => ( ! [Y: A] :
                ( ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X7) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_5439_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X7: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X7))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X7) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5440_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X7: set(A),A2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3)) )
         => ( ! [Y: A] :
                ( ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A2)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X7) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_5441_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X7: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X7))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X7) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5442_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),Z: A] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X7)),Z)) ) ) ) ).

% cSup_least
tff(fact_5443_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),A2: A] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
           => ( ! [Y: A] :
                  ( ! [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X7))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X7) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5444_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X7))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X7))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X7))) ) ) ) ).

% le_cSup_finite
tff(fact_5445_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),Z: A] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X7))) ) ) ) ).

% cInf_greatest
tff(fact_5446_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),A2: A] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3)) )
           => ( ! [Y: A] :
                  ( ! [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X7))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X4)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A2)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X7) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5447_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X7: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X7))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X7))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X7)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_5448_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A2)) ) ) ) ).

% cSup_abs_le
tff(fact_5449_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Inf_le_Sup
tff(fact_5450_subset__Pow__Union,axiom,
    ! [A: $tType,A3: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A3),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)))) ).

% subset_Pow_Union
tff(fact_5451_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(A)] :
      ( ! [X8: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X8),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),B4)) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B4)) ) ) ).

% Inter_subset
tff(fact_5452_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [Y: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
         => ( ! [Y: A] :
                ( ! [Z3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),Y)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = X ) ) ) ) ).

% Sup_eqI
tff(fact_5453_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ! [A4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X4)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ).

% Sup_mono
tff(fact_5454_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ).

% Sup_least
tff(fact_5455_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Sup_upper
tff(fact_5456_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_5457_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% Sup_upper2
tff(fact_5458_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I3)) )
         => ( ! [Y: A] :
                ( ! [I4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),I4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A3) = X ) ) ) ) ).

% Inf_eqI
tff(fact_5459_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B4))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B3)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ).

% Inf_mono
tff(fact_5460_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ).

% Inf_lower
tff(fact_5461_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V)) ) ) ) ).

% Inf_lower2
tff(fact_5462_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X5)) ) ) ) ).

% le_Inf_iff
tff(fact_5463_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ).

% Inf_greatest
tff(fact_5464_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
         => ? [Y3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y3),B4))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Y3)) ) )
     => 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)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4))) ) ).

% Union_subsetI
tff(fact_5465_Union__upper,axiom,
    ! [A: $tType,B4: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B4),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ) ).

% Union_upper
tff(fact_5466_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X8: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X8),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),C3)) )
     => 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)),A3)),C3)) ) ).

% Union_least
tff(fact_5467_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X8: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X8),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),X8)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3))) ) ).

% Inter_greatest
tff(fact_5468_Inter__lower,axiom,
    ! [A: $tType,B4: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B4),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B4)) ) ).

% Inter_lower
tff(fact_5469_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3)))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X5)) ) ) ) ) ).

% le_Sup_iff
tff(fact_5470_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: 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),A3)),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5)) ) ) ) ) ).

% Inf_le_iff
tff(fact_5471_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V4)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% less_eq_Sup
tff(fact_5472_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ).

% Sup_subset_mono
tff(fact_5473_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V4),U)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_5474_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ).

% Inf_superset_mono
tff(fact_5475_Union__mono,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A3),B4))
     => 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)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4))) ) ).

% Union_mono
tff(fact_5476_Inter__anti__mono,axiom,
    ! [A: $tType,B4: set(set(A)),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B4),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4))) ) ).

% Inter_anti_mono
tff(fact_5477_card__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( pp(aa(set(set(A)),bool,finite_finite(set(A)),C3))
     => ( pp(aa(set(A),bool,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))
       => ( ! [C4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
             => ( aa(set(A),nat,finite_card(A),C4) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C1),C3))
               => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C22),C3))
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C3)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ) ) ).

% card_partition
tff(fact_5478_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_mm(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_5479_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_5480_Int__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% Int_iff
tff(fact_5481_IntI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4))
       => 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)),inf_inf(set(A)),A3),B4))) ) ) ).

% IntI
tff(fact_5482_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.bounded_iff
tff(fact_5483_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: 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),Y4),Z)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% le_inf_iff
tff(fact_5484_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),B4)) ) ) ).

% Int_subset_iff
tff(fact_5485_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_5486_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ).

% Int_insert_right_if0
tff(fact_5487_insert__inter__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),A3)),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) ).

% insert_inter_insert
tff(fact_5488_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),B4)),C3) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_5489_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3) ) ) ).

% Int_insert_left_if0
tff(fact_5490_Pow__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A3)),pow2(A,B4)) ).

% Pow_Int_eq
tff(fact_5491_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),A3)),B4) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_5492_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),A3)),B4) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_5493_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B4: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(A),set(A),insert(A,A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),A3) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_5494_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,B2),B4)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_5495_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_5496_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_5497_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_5498_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_5499_Compl__disjoint,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_5500_Compl__disjoint2,axiom,
    ! [A: $tType,A3: 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)),A3)),A3) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_5501_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ).

% Diff_Compl
tff(fact_5502_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),F2: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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_mn(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P)),A3) = 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)),A3),collect(B,P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_5503_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),P: fun(B,bool),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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_mo(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F2)),A3) = 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)),A3),collect(B,P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_5504_sum__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_mp(fun(B,bool),fun(B,A),P)),A3) = 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)),A3),collect(B,P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_5505_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)))) ) ).

% Sup_inter_less_eq
tff(fact_5506_Union__Int__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(set(A))] : 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)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)))) ).

% Union_Int_subset
tff(fact_5507_Int__insert__right,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ) ) ) ).

% Int_insert_right
tff(fact_5508_Int__insert__left,axiom,
    ! [A: $tType,A2: A,C3: set(A),B4: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),B4)),C3) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert(A,A2),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3) ) ) ) ).

% Int_insert_left
tff(fact_5509_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: 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),A2),B2)),C2)) ) ) ).

% inf.coboundedI2
tff(fact_5510_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),C2)) ) ) ).

% inf.coboundedI1
tff(fact_5511_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_5512_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_5513_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: 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),A2),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_5514_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: 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),A2),B2)),A2)) ) ).

% inf.cobounded1
tff(fact_5515_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_5516_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( 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),Y4),Z))) ) ) ) ).

% inf_greatest
tff(fact_5517_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).

% inf.boundedI
tff(fact_5518_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.boundedE
tff(fact_5519_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y4) = Y4 ) ) ) ).

% inf_absorb2
tff(fact_5520_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y4) = X ) ) ) ).

% inf_absorb1
tff(fact_5521_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_5522_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_5523_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y4) = X ) ) ) ).

% le_iff_inf
tff(fact_5524_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y4: A] :
          ( ! [X3: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y)),X3))
         => ( ! [X3: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y)),Y))
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F2,Y),Z2))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y4) = aa(A,A,aa(A,fun(A,A),F2,X),Y4) ) ) ) ) ) ).

% inf_unique
tff(fact_5525_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% inf.orderI
tff(fact_5526_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_5527_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: 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),A2),B2)),X)) ) ) ).

% le_infI2
tff(fact_5528_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),X)) ) ) ).

% le_infI1
tff(fact_5529_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2))) ) ) ) ).

% inf_mono
tff(fact_5530_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( 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),A2),B2))) ) ) ) ).

% le_infI
tff(fact_5531_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: 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),A2),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_5532_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: 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),Y4)),Y4)) ) ).

% inf_le2
tff(fact_5533_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y4: 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),Y4)),X)) ) ).

% inf_le1
tff(fact_5534_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y4: 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),Y4)),X)) ) ).

% inf_sup_ord(1)
tff(fact_5535_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y4: 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),Y4)),Y4)) ) ).

% inf_sup_ord(2)
tff(fact_5536_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),D4))
       => 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)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D4))) ) ) ).

% Int_mono
tff(fact_5537_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)),A3)) ).

% Int_lower1
tff(fact_5538_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)),B4)) ).

% Int_lower2
tff(fact_5539_Int__absorb1,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_5540_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = A3 ) ) ).

% Int_absorb2
tff(fact_5541_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),B4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ).

% Int_greatest
tff(fact_5542_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => ( pp(aa(A,bool,P,X3))
             => pp(aa(A,bool,Q,X3)) ) )
       => pp(aa(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)),inf_inf(set(A)),A3),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),collect(A,Q)))) ) ) ).

% Int_Collect_mono
tff(fact_5543_Int__left__commute,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_left_commute
tff(fact_5544_Int__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ).

% Int_left_absorb
tff(fact_5545_Int__commute,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),A3) ).

% Int_commute
tff(fact_5546_Int__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),A3) = A3 ).

% Int_absorb
tff(fact_5547_Int__assoc,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) ).

% Int_assoc
tff(fact_5548_IntD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ).

% IntD2
tff(fact_5549_IntD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% IntD1
tff(fact_5550_IntE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),inf_inf(set(A)),A3),B4)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% IntE
tff(fact_5551_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_mq(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)),inf_inf(set(A)),collect(A,P)),collect(A,Q)) ).

% Collect_conj_eq
tff(fact_5552_Int__Collect,axiom,
    ! [A: $tType,X: A,A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & pp(aa(A,bool,P,X)) ) ) ).

% Int_Collect
tff(fact_5553_Int__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_mr(set(A),fun(set(A),fun(A,bool)),A3),B4)) ).

% Int_def
tff(fact_5554_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) ).

% Diff_Int_distrib2
tff(fact_5555_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B4)) ).

% Diff_Int_distrib
tff(fact_5556_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ).

% Diff_Diff_Int
tff(fact_5557_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),B4) ).

% Diff_Int2
tff(fact_5558_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C3)) ).

% Int_Diff
tff(fact_5559_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y4)) ) ).

% diff_eq
tff(fact_5560_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_5561_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4)) ) ) ).

% disjoint_iff
tff(fact_5562_Int__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B4) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_5563_Int__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_5564_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
         => ! [Xa4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),B4))
             => ( X5 != Xa4 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_5565_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_5566_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = A3 ) ) ).

% Diff_triv
tff(fact_5567_Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Diff_eq
tff(fact_5568_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y))
           => ( 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)),A3) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5569_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y4) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y4))) ) ) ).

% inf_shunt
tff(fact_5570_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B4))) ) ).

% disjoint_eq_subset_Compl
tff(fact_5571_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_5572_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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)),A3),B4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_ms(fun(B,A),fun(set(B),fun(B,A)),G),B4)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_5573_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( 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)),S),T6)))
                   => ( aa(B,A,G,I3) = zero_zero(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T6)))
                     => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T6) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5574_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = 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)),A3),B4))),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)),A3),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_5575_prod_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4))) ) ) ) ).

% prod.Int_Diff
tff(fact_5576_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
                 => ( 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)),S),T6)))
                   => ( aa(B,A,G,I3) = one_one(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T6)))
                     => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
                 => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(B,A,H,T6) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_5577_card__Diff__subset__Int,axiom,
    ! [A: $tType,A3: set(A),B4: 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)),A3),B4)))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ).

% card_Diff_subset_Int
tff(fact_5578_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_5579_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_5580_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_5581_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mt(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),collect(B,P)))),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).

% prod.If_cases
tff(fact_5582_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X3))) )
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C3))
             => ! [Xa3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa3),C3))
                 => ( ( X3 != Xa3 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa3) = bot_bot(set(A)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))) ) ) ) ).

% dvd_partition
tff(fact_5583_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: set(B),F2: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( 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),A3)),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_mu(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)),A3),collect(B,aa(A,fun(B,bool),aTP_Lamp_mv(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)),A3),collect(B,aa(A,fun(B,bool),aTP_Lamp_mw(fun(B,A),fun(A,fun(B,bool)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5584_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_5585_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_5586_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: 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),A2),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_mx(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_5587_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_5588_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_5589_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_5590_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_5591_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_5592_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_5593_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ).

% inf_set_def
tff(fact_5594_funpow__add,axiom,
    ! [A: $tType,M: 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),M),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)),M),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_5595_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_5596_funpow__mod__eq,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A),X: A,M: nat] :
      ( ( 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) = X )
     => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),modulo_modulo(nat,M,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)),M),F2),X) ) ) ).

% funpow_mod_eq
tff(fact_5597_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S: set(A),N: nat] :
      ( bij_betw(A,A,F2,S,S)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),S,S) ) ).

% bij_betw_funpow
tff(fact_5598_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_5599_funpow__mult,axiom,
    ! [A: $tType,N: nat,M: 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)),M),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),M),N)),F2) ).

% funpow_mult
tff(fact_5600_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F2: fun(A,nat),X: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F2,X)),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F2,X))) ) ).

% funpow_times_power
tff(fact_5601_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_5602_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,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,M,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5603_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,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,M,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_5604_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5605_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_5606_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(fun(nat,nat),fun(nat,bool),comp(nat,bool,nat,bit_se5641148757651400278ts_bit(A,A2)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_5607_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_my(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_5608_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5935395276787703475ssThan(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_5609_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_5610_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_5611_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y4: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_5612_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y4: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y4)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_5613_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y4: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y4),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_5614_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y4: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y4),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_5615_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y4: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y4),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_5616_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y4: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y4),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_5617_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_5618_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_5619_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_5620_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,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_5621_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,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_5622_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,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),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_5623_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_5624_sum_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(set(B)),G: fun(B,A)] :
          ( ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X3),B4))
             => pp(aa(set(B),bool,finite_finite(B),X3)) )
         => ( ! [A13: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A13),B4))
               => ! [A24: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A24),B4))
                   => ( ( A13 != A24 )
                     => ! [X3: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A13))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A24))
                           => ( aa(B,A,G,X3) = 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)),B4)) = 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),B4) ) ) ) ) ).

% sum.Union_comp
tff(fact_5625_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: 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_mg(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),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,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5626_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: 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_mg(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),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,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5627_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_mg(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5628_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_mg(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5629_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,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,M,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_5630_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_5631_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : groups7121269368397514597t_prod(int,A,G,set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_5632_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,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,M,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_5633_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,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),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_5634_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_5635_scale__left__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y4: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_5636_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : groups7121269368397514597t_prod(int,A,G,set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_5637_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y4: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_5638_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_5639_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y4: 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),Y4))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = X )
          & ( aa(nat,A,F7,N) = Y4 )
          & ! [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,F7,I2)),aa(nat,A,F7,aa(nat,nat,suc,I2)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_5640_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_5641_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))
     => ~ ! [Y: 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),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y),Z)) ) ) ).

% relpowp_Suc_E
tff(fact_5642_relpowp__Suc__I,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y4: 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),Y4))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),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_5643_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))
     => ? [Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
          & 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),Y),Z)) ) ) ).

% relpowp_Suc_D2
tff(fact_5644_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))
     => ~ ! [Y: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
           => ~ 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),Y),Z)) ) ) ).

% relpowp_Suc_E2
tff(fact_5645_relpowp__Suc__I2,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y4: A,N: nat,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
     => ( 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),Y4),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_5646_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R: 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)),R) = fequal(A) ).

% relpowp.simps(1)
tff(fact_5647_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y4: 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),Y4))
     => ( X = Y4 ) ) ).

% relpowp_0_E
tff(fact_5648_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_5649_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 ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
               => ~ 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))),M4),P),Y),Z)) ) ) ) ) ).

% relpowp_E2
tff(fact_5650_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 ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( 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))),M4),P),X),Y))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y),Z)) ) ) ) ) ).

% relpowp_E
tff(fact_5651_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_5652_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: 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),J),aa(nat,nat,suc,I))))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5653_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_5654_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_kp(nat,fun(nat,bool)),aTP_Lamp_ec(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5655_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),code_nat_of_integer(K)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).

% of_nat_of_integer
tff(fact_5656_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y4: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y4)) ).

% bot2E
tff(fact_5657_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5658_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_5659_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_5660_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5661_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_mz(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5662_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5663_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_na(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_5664_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),insert(A,X),A3)) = linorder_insort_key(A,A,aTP_Lamp_nb(A,A),X,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5665_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_5666_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_5667_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_nb(A,A),X,Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_5668_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_5669_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linorder_insort_key(B,A,F2,X,Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs)) ) ) ).

% sorted_insort_key
tff(fact_5670_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( linorder_insort_key(A,A,aTP_Lamp_nb(A,A),A2,remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_5671_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( linord4507533701916653071of_set(A,A3) = linorder_insort_key(A,A,aTP_Lamp_nb(A,A),X,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5672_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)))
       => ( code_int_of_integer(K) = aa(int,int,uminus_uminus(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) )
           => ( code_int_of_integer(K) = zero_zero(int) ) )
          & ( ( K != zero_zero(code_integer) )
           => ( code_int_of_integer(K) = aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_nc(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_5673_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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% times_int.abs_eq
tff(fact_5674_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y4: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Y4))) ) ) ) ) ).

% arg_min_least
tff(fact_5675_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_5676_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_5677_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% times_integer.rep_eq
tff(fact_5678_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : 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),code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% minus_integer.rep_eq
tff(fact_5679_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : 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),code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% divide_integer.rep_eq
tff(fact_5680_int__of__integer__of__nat,axiom,
    ! [N: nat] : 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_5681_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y)) ).

% eq_Abs_Integ
tff(fact_5682_int_Oabs__induct,axiom,
    ! [P: fun(int,bool),X: int] :
      ( ! [Y: product_prod(nat,nat)] : pp(aa(int,bool,P,aa(product_prod(nat,nat),int,abs_Integ,Y)))
     => pp(aa(int,bool,P,X)) ) ).

% int.abs_induct
tff(fact_5683_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),code_int_of_integer(X)),code_int_of_integer(Xa))) ) ).

% less_eq_integer.rep_eq
tff(fact_5684_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_eq_iff
tff(fact_5685_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_5686_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),X),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),X),code_int_of_integer(Xa)) ).

% unset_bit_integer.rep_eq
tff(fact_5687_set__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),X),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),X),code_int_of_integer(Xa)) ).

% set_bit_integer.rep_eq
tff(fact_5688_flip__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),X),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),X),code_int_of_integer(Xa)) ).

% flip_bit_integer.rep_eq
tff(fact_5689_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_5690_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_5691_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_nf(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_5692_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_5693_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_ng(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_5694_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_ni(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_5695_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_nk(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_eq_int.abs_eq
tff(fact_5696_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_nm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% plus_int.abs_eq
tff(fact_5697_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_no(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% minus_int.abs_eq
tff(fact_5698_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_5699_insert__subsetI,axiom,
    ! [A: $tType,X: A,A3: set(A),X7: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),A3))
       => 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),X7)),A3)) ) ) ).

% insert_subsetI
tff(fact_5700_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: 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),J),I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5701_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_5702_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5703_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_5704_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_5705_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Gcd(A,A3) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_5706_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2)) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5707_Gcd__dvd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,A3)),A2)) ) ) ).

% Gcd_dvd
tff(fact_5708_dvd__GcdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A3: set(A),Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),gcd_Gcd(A,A3)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4)) ) ) ) ).

% dvd_GcdD
tff(fact_5709_dvd__Gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),gcd_Gcd(A,A3)))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),X5)) ) ) ) ).

% dvd_Gcd_iff
tff(fact_5710_Gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),A2: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),gcd_Gcd(A,A3))) ) ) ).

% Gcd_greatest
tff(fact_5711_Gcd__dvd__nat,axiom,
    ! [A2: nat,A3: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),A2),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),gcd_Gcd(nat,A3)),A2)) ) ).

% Gcd_dvd_nat
tff(fact_5712_Gcd__greatest__nat,axiom,
    ! [A3: set(nat),A2: nat] :
      ( ! [B3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),B3),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),B3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),gcd_Gcd(nat,A3))) ) ).

% Gcd_greatest_nat
tff(fact_5713_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_5714_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: 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,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5715_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_5716_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | 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),B2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),A2)) ) ) ) ).

% Ioc_disjoint
tff(fact_5717_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or3652927894154168847AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_5718_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_5719_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_5720_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5721_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5722_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: 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,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5723_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5724_prop__restrict,axiom,
    ! [A: $tType,X: A,Z6: set(A),X7: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Z6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z6),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),X7),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_5725_Collect__restrict,axiom,
    ! [A: $tType,X7: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(set(A),fun(fun(A,bool),fun(A,bool)),X7),P))),X7)) ).

% Collect_restrict
tff(fact_5726_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_5727_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_nk(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_5728_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_ni(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_5729_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,collect(nat,aTP_Lamp_np(nat,bool))) ) ).

% semiring_char_def
tff(fact_5730_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_5731_Gcd__dvd__int,axiom,
    ! [A2: int,A3: set(int)] :
      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),A2),A3))
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),gcd_Gcd(int,A3)),A2)) ) ).

% Gcd_dvd_int
tff(fact_5732_Gcd__greatest__int,axiom,
    ! [A3: set(int),A2: int] :
      ( ! [B3: int] :
          ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),B3),A3))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A2),B3)) )
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A2),gcd_Gcd(int,A3))) ) ).

% Gcd_greatest_int
tff(fact_5733_Gcd__int__greater__eq__0,axiom,
    ! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5))) ).

% Gcd_int_greater_eq_0
tff(fact_5734_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_5735_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_ng(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_5736_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_nf(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_5737_prod__encode__def,axiom,
    nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_nq(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5738_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_5739_prod__encode__eq,axiom,
    ! [X: product_prod(nat,nat),Y4: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,X) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y4) )
    <=> ( X = Y4 ) ) ).

% prod_encode_eq
tff(fact_5740_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_5741_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5742_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),B2)))) ).

% le_prod_encode_1
tff(fact_5743_le__prod__encode__2,axiom,
    ! [B2: nat,A2: 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,A2),B2)))) ).

% le_prod_encode_2
tff(fact_5744_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_5745_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_5746_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_5747_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_5748_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( 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),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5749_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_5750_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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_5751_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_no(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_5752_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_nm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_5753_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y4: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y4)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y4))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_5754_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_5755_pred__nat__def,axiom,
    pred_nat = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_nr(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_5756_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_5757_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_5758_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5759_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5760_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5761_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Y4: A] :
          ( ( X = Y4 )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) ) ) ).

% eq_iff_iszero_diff
tff(fact_5762_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y4: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y4) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y4)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_5763_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_5764_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_5765_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_5766_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y4: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(num,A,numeral_numeral(A),Y4) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,Y4)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_5767_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_5768_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y4: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y4)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y4)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_5769_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_5770_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y4: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y4) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y4))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_5771_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y4: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y4)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y4))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_5772_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y4: 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),Y4)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y4,X)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_5773_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_5774_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y4: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y4) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y4)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_5775_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P3: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_jz(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),insert(B,I),I5)) = groups1962203154675924110t_prod(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),insert(B,I),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P3,I)),groups1962203154675924110t_prod(B,A,P3,I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_5776_pow_Osimps_I3_J,axiom,
    ! [X: num,Y4: num] : pow(X,aa(num,num,bit1,Y4)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y4))),X) ).

% pow.simps(3)
tff(fact_5777_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y4: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y4))
       => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_ns(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y4)) = 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),Y4),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y4))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y4))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_ns(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y4)) = 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),Y4))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_ns(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y4)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5778_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B,A3: set(B)] :
      ( ( B2 = aa(B,A,F2,X) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),A3))) ) ) ).

% image_eqI
tff(fact_5779_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_nt(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_5780_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_5781_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F2),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_5782_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_5783_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B4: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),insert(B,A2),B4)) = aa(set(A),set(A),insert(A,aa(B,A,F2,A2)),aa(set(B),set(A),image(B,A,F2),B4)) ).

% image_insert
tff(fact_5784_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(B),set(B),insert(B,aa(A,B,F2,X)),aa(set(A),set(B),image(A,B,F2),A3)) = aa(set(A),set(B),image(A,B,F2),A3) ) ) ).

% insert_image
tff(fact_5785_bij__betw__Suc,axiom,
    ! [M10: set(nat),N5: set(nat)] :
      ( bij_betw(nat,nat,suc,M10,N5)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M10) = N5 ) ) ).

% bij_betw_Suc
tff(fact_5786_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_5787_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_5788_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_5789_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_5790_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N5: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N5,A3)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N5) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_5791_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_nu(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5792_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5793_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_5794_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5795_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_nv(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5796_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),B4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),collect(A,P))),B4)) ) ).

% image_Collect_subsetI
tff(fact_5797_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B4: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B4)))) ).

% image_Int_subset
tff(fact_5798_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_5799_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5800_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3))) ).

% zero_notin_Suc_image
tff(fact_5801_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B10: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite(A),B10))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image(B,A,F2),A3))) )
         => pp(aa(set(A),bool,P,B10)) )
    <=> ! [B10: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),B10))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B10))) ) ) ).

% all_finite_subset_image
tff(fact_5802_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ? [B10: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),B10))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image(B,A,F2),A3)))
          & pp(aa(set(A),bool,P,B10)) )
    <=> ? [B10: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),B10))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B10))) ) ) ).

% ex_finite_subset_image
tff(fact_5803_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),A3)))
       => ? [C8: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),A3))
            & pp(aa(set(B),bool,finite_finite(B),C8))
            & ( B4 = aa(set(B),set(A),image(B,A,F2),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_5804_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F2),A3)))
       => pp(aa(set(B),bool,finite_finite(B),B4)) ) ) ).

% finite_surj
tff(fact_5805_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),A3)))
     => ~ ! [X3: B] :
            ( ( B2 = aa(B,A,F2,X3) )
           => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3)) ) ) ).

% imageE
tff(fact_5806_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_nw(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_5807_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_nx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F2),A3),P)) = aa(set(B),set(A),image(B,A,F2),collect(B,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ny(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_5808_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X)),aa(set(A),set(B),image(A,B,F2),A3))) ) ).

% imageI
tff(fact_5809_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(set(B),set(A),image(B,A,F2),A3)))
    <=> ? [X5: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
          & ( Z = aa(B,A,F2,X5) ) ) ) ).

% image_iff
tff(fact_5810_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(B),set(A),image(B,A,F2),A3)))
          & pp(aa(A,bool,P,X4)) )
     => ? [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
          & pp(aa(A,bool,P,aa(B,A,F2,X3))) ) ) ).

% bex_imageD
tff(fact_5811_image__cong,axiom,
    ! [B: $tType,A: $tType,M10: set(A),N5: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M10 = N5 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),N5))
           => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
       => ( aa(set(A),set(B),image(A,B,F2),M10) = aa(set(A),set(B),image(A,B,G),N5) ) ) ) ).

% image_cong
tff(fact_5812_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(set(B),set(A),image(B,A,F2),A3)))
         => pp(aa(A,bool,P,X3)) )
     => ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
         => pp(aa(A,bool,P,aa(B,A,F2,X4))) ) ) ).

% ball_imageD
tff(fact_5813_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),B2: B,F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( ( B2 = aa(A,B,F2,X) )
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% rev_image_eqI
tff(fact_5814_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),A3)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A3))
          & ( B4 = aa(set(B),set(A),image(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_5815_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B4))
    <=> ! [X5: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X5)),B4)) ) ) ).

% image_subset_iff
tff(fact_5816_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),A3)))
     => ~ ! [C8: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),A3))
           => ( B4 != aa(set(B),set(A),image(B,A,F2),C8) ) ) ) ).

% subset_imageE
tff(fact_5817_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),B4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4)) ) ).

% image_subsetI
tff(fact_5818_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4))) ) ).

% image_mono
tff(fact_5819_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B10: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image(B,A,F2),A3)))
         => pp(aa(set(A),bool,P,B10)) )
    <=> ! [B10: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3))
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B10))) ) ) ).

% all_subset_image
tff(fact_5820_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A3) = B4 )
     => ( aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),pow2(B,A3)) = pow2(A,B4) ) ) ).

% image_Pow_surj
tff(fact_5821_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),pow2(B,A3))),pow2(A,B4))) ) ).

% image_Pow_mono
tff(fact_5822_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B4: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => ? [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),aa(C,A,G,X4))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B4))
               => ? [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,J2)),aa(B,A,F2,X4))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B4)) ) ) ) ) ).

% SUP_eq
tff(fact_5823_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B4: set(C),G: fun(C,A),F2: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => ? [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,X4)),aa(B,A,F2,I3))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B4))
               => ? [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(C,A,G,J2))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,G),B4)) ) ) ) ) ).

% INF_eq
tff(fact_5824_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B4: 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)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B4))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4)))) ).

% image_diff_subset
tff(fact_5825_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A),X: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),X)) )
         => ( ! [Y: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I4)),Y)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_5826_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B4: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( ! [N2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
             => ? [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,N2)),aa(C,A,G,X4))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B4)))) ) ) ).

% SUP_mono
tff(fact_5827_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U)) ) ) ).

% SUP_least
tff(fact_5828_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),A3)))) ) ) ).

% SUP_mono'
tff(fact_5829_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ).

% SUP_upper
tff(fact_5830_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U))
        <=> ! [X5: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X5)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_5831_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),U: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ) ).

% SUP_upper2
tff(fact_5832_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),X: A,F2: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F2,I3))) )
         => ( ! [Y: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(B,A,F2,I4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)) = X ) ) ) ) ).

% INF_eqI
tff(fact_5833_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A3: set(C),F2: fun(C,A),G: fun(B,A)] :
          ( ! [M4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M4),B4))
             => ? [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F2,X4)),aa(B,A,G,M4))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B4)))) ) ) ).

% INF_mono
tff(fact_5834_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(B,A,F2,I))) ) ) ).

% INF_lower
tff(fact_5835_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),A3)))) ) ) ).

% INF_mono'
tff(fact_5836_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),U)) ) ) ) ).

% INF_lower2
tff(fact_5837_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))))
        <=> ! [X5: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X5))) ) ) ) ).

% le_INF_iff
tff(fact_5838_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ).

% INF_greatest
tff(fact_5839_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),C2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_nz(B,fun(A,B),C2)),A3) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_5840_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A3: set(B),C2: A] :
      ( ( ( A3 = bot_bot(set(B)) )
       => ( aa(set(B),set(A),image(B,A,aTP_Lamp_oa(A,fun(B,A),C2)),A3) = bot_bot(set(A)) ) )
      & ( ( A3 != bot_bot(set(B)) )
       => ( aa(set(B),set(A),image(B,A,aTP_Lamp_oa(A,fun(B,A),C2)),A3) = aa(set(A),set(A),insert(A,C2),bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_5841_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_5842_Gcd__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,X3)),aa(B,A,G,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,aa(set(B),set(A),image(B,A,F2),A3))),gcd_Gcd(A,aa(set(B),set(A),image(B,A,G),A3)))) ) ) ).

% Gcd_mono
tff(fact_5843_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
           => ( aa(A,B,F2,Y) = aa(A,B,F2,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F2),A3)) = aa(A,B,F2,X) ) ) ) ).

% the_elem_image_unique
tff(fact_5844_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F2: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),aa(B,A,F2,X5))) ) ) ) ) ).

% le_SUP_iff
tff(fact_5845_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: 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),aa(set(B),set(A),image(B,A,F2),A3))),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X5)),Y5)) ) ) ) ) ).

% INF_le_iff
tff(fact_5846_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),C2: A,F2: fun(B,A)] :
          ( ( 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_eq(A),C2),aa(B,A,F2,I3))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),I5)) = C2 )
            <=> ! [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),I5))
                 => ( aa(B,A,F2,X5) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5847_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),M10: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),M10)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),M10)) ) ) ) ).

% cSUP_least
tff(fact_5848_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F2: fun(B,A),C2: A] :
          ( ( 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_eq(A),aa(B,A,F2,I3)),C2)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),I5)) = C2 )
            <=> ! [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),I5))
                 => ( aa(B,A,F2,X5) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5849_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),M: A,F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F2,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ) ).

% cINF_greatest
tff(fact_5850_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(A),nat,finite_card(A),A3))) ) ).

% card_image_le
tff(fact_5851_prod_Odistrib__triv_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),I5))
         => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gt(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_5852_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),B4)))) ) ) ) ).

% SUP_subset_mono
tff(fact_5853_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B4)))) ) ) ) ).

% INF_superset_mono
tff(fact_5854_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T6: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( pp(aa(set(C),bool,finite_finite(C),T6))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,G),S)),T6))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_oc(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G),H)),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S) ) ) ) ) ) ).

% sum.group
tff(fact_5855_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T6: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( pp(aa(set(C),bool,finite_finite(C),T6))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,G),S)),T6))
             => ( groups7121269368397514597t_prod(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_od(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G),H),T6) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.group
tff(fact_5856_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).

% numeral_sqr
tff(fact_5857_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T6: 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)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G,T6) = groups1962203154675924110t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_5858_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T6: 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)),S),T6))
         => ( ! [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)),T6),S)))
               => ( aa(B,A,H,I3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G,S) = groups1962203154675924110t_prod(B,A,H,T6) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_5859_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,T6) = groups1962203154675924110t_prod(B,A,G,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_5860_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T6),S)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,S) = groups1962203154675924110t_prod(B,A,G,T6) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_5861_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [X3: B,Y: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),A3))
                 => ( ( X3 != Y )
                   => ( ( aa(B,C,H,X3) = aa(B,C,H,Y) )
                     => ( aa(C,A,G,aa(B,C,H,X3)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image(B,C,H),A3)) = 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)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5862_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ).

% INF_le_SUP
tff(fact_5863_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F2),A3)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% surj_card_le
tff(fact_5864_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y4: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y4)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y4)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5865_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_5866_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_5867_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)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5868_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)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5869_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)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5870_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)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_5871_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_jz(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_jz(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gt(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_5872_pow_Osimps_I2_J,axiom,
    ! [X: num,Y4: num] : pow(X,aa(num,num,bit0,Y4)) = sqr(pow(X,Y4)) ).

% pow.simps(2)
tff(fact_5873_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),aa(set(C),set(A),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_5874_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = 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),Y4)) ) )
          & ( ~ 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),Y4))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y4),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),Y4))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5875_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_5876_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = 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),Y4),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y4),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),Y4))
           => ( aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y4)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5877_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_of(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_of(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_of(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5878_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_og(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_og(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,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),M),A2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_og(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,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),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5879_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5880_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5881_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( pp(aa(set(B),bool,finite_finite(B),R))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),R))
             => ( 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_oj(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S) = 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_ol(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5882_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_om(A,fun(nat,A),B4)),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B4) ) ).

% INF_nat_binary
tff(fact_5883_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F8: fun(A,B),A8: set(A),A9: set(B),F2: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F8,A8,A9)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),A3)),A8))
       => ( bij_betw(C,A,F2,A3,A8)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F8),F2),A3,A9) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_5884_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_Compl
tff(fact_5885_Gcd__int__eq,axiom,
    ! [N5: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N5)) ).

% Gcd_int_eq
tff(fact_5886_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_oo(fun(B,set(A)),fun(B,set(set(A))),B4)),A3))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))))) ).

% UN_Pow_subset
tff(fact_5887_Inf__real__def,axiom,
    ! [X7: set(real)] : aa(set(real),real,complete_Inf_Inf(real),X7) = aa(real,real,uminus_uminus(real),aa(set(real),real,complete_Sup_Sup(real),aa(set(real),set(real),image(real,real,uminus_uminus(real)),X7))) ).

% Inf_real_def
tff(fact_5888_finite__int__iff__bounded,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S)),set_ord_lessThan(int,K3))) ) ).

% finite_int_iff_bounded
tff(fact_5889_finite__int__iff__bounded__le,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S)),set_ord_atMost(int,K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_5890_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I5: set(B),B4: set(A)] :
      ( 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)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),B4))
    <=> ! [X5: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A3,X5)),B4)) ) ) ).

% UN_subset_iff
tff(fact_5891_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3)))) ) ).

% UN_upper
tff(fact_5892_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(A,set(B)),C3: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X3)),C3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),C3)) ) ).

% UN_least
tff(fact_5893_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),B4)))) ) ) ).

% UN_mono
tff(fact_5894_UN__extend__simps_I6_J,axiom,
    ! [L5: $tType,K9: $tType,A3: fun(K9,set(L5)),C3: set(K9),B4: 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)),aa(set(K9),set(set(L5)),image(K9,set(L5),A3),C3))),B4) = aa(set(set(L5)),set(L5),complete_Sup_Sup(set(L5)),aa(set(K9),set(set(L5)),image(K9,set(L5),aa(set(L5),fun(K9,set(L5)),aTP_Lamp_op(fun(K9,set(L5)),fun(set(L5),fun(K9,set(L5))),A3),B4)),C3)) ).

% UN_extend_simps(6)
tff(fact_5895_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))))
    <=> ! [X5: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(B,set(A),A3,X5))) ) ) ).

% INT_subset_iff
tff(fact_5896_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),B4))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),A3)))) ) ) ).

% INT_anti_mono
tff(fact_5897_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(B),B4: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),aa(A,set(B),B4,X3))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3)))) ) ).

% INT_greatest
tff(fact_5898_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),aa(A,set(B),B4,A2))) ) ).

% INT_lower
tff(fact_5899_in__image__insert__iff,axiom,
    ! [A: $tType,B4: set(set(A)),X: A,A3: set(A)] :
      ( ! [C8: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C8),B4))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C8)) )
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,X)),B4)))
      <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
          & 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B4)) ) ) ) ).

% in_image_insert_iff
tff(fact_5900_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_5901_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_5902_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C),A8: fun(A,set(C))] :
      ( ! [I3: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I3)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
           => bij_betw(B,C,F2,aa(A,set(B),A3,I3),aa(A,set(C),A8,I3)) )
       => bij_betw(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),A8),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_5903_UN__extend__simps_I7_J,axiom,
    ! [M11: $tType,N10: $tType,A3: set(M11),B4: fun(N10,set(M11)),C3: set(N10)] : aa(set(M11),set(M11),aa(set(M11),fun(set(M11),set(M11)),minus_minus(set(M11)),A3),aa(set(set(M11)),set(M11),complete_Inf_Inf(set(M11)),aa(set(N10),set(set(M11)),image(N10,set(M11),B4),C3))) = aa(set(set(M11)),set(M11),complete_Sup_Sup(set(M11)),aa(set(N10),set(set(M11)),image(N10,set(M11),aa(fun(N10,set(M11)),fun(N10,set(M11)),aTP_Lamp_oq(set(M11),fun(fun(N10,set(M11)),fun(N10,set(M11))),A3),B4)),C3)) ).

% UN_extend_simps(7)
tff(fact_5904_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X7: set(A),A3: set(product_prod(A,B)),Y6: set(B),P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ( X7 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A3) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X7))
             => ! [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),Y6))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Xa3))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Xa3)) ) ) )
         => ( 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))),A3),collect(product_prod(A,B),product_case_prod(A,B,bool,P))))
           => 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))),A3),collect(product_prod(A,B),product_case_prod(A,B,bool,Q)))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_5905_INT__extend__simps_I4_J,axiom,
    ! [G3: $tType,H4: $tType,C3: set(H4),A3: set(G3),B4: fun(H4,set(G3))] :
      ( ( ( C3 = bot_bot(set(H4)) )
       => ( aa(set(G3),set(G3),aa(set(G3),fun(set(G3),set(G3)),minus_minus(set(G3)),A3),aa(set(set(G3)),set(G3),complete_Sup_Sup(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),B4),C3))) = A3 ) )
      & ( ( C3 != bot_bot(set(H4)) )
       => ( aa(set(G3),set(G3),aa(set(G3),fun(set(G3),set(G3)),minus_minus(set(G3)),A3),aa(set(set(G3)),set(G3),complete_Sup_Sup(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),B4),C3))) = aa(set(set(G3)),set(G3),complete_Inf_Inf(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),aa(fun(H4,set(G3)),fun(H4,set(G3)),aTP_Lamp_or(set(G3),fun(fun(H4,set(G3)),fun(H4,set(G3))),A3),B4)),C3)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_5906_subset__subseqs,axiom,
    ! [A: $tType,X7: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))) ) ).

% subset_subseqs
tff(fact_5907_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_os(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_5908_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite(A),I5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ot(fun(A,set(B)),fun(A,nat),A3)),I5))) ) ).

% card_UN_le
tff(fact_5909_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_5910_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) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_5911_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S2)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T2)) ) ).

% translation_diff
tff(fact_5912_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F8: fun(B,A),F2: fun(A,B),A8: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ( aa(B,A,F8,aa(A,B,F2,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A8))
           => ( aa(A,B,F2,aa(B,A,F8,X3)) = X3 ) )
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),A8))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F8),A8)),A3))
           => bij_betw(A,B,F2,A3,A8) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_5913_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A8: set(B),B4: set(A),B12: set(B)] :
      ( bij_betw(A,B,F2,A3,A8)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
       => ( ( aa(set(A),set(B),image(A,B,F2),B4) = B12 )
         => bij_betw(A,B,F2,B4,B12) ) ) ) ).

% bij_betw_subset
tff(fact_5914_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),S2)),aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_Int
tff(fact_5915_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),S2)),aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),T2)) ) ).

% translation_subtract_diff
tff(fact_5916_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),X: C,X7: set(A)] :
      ( 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)),aa(set(B),set(set(A)),image(B,set(A),F2),aa(C,set(B),G,X)))),X7))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G,X)),collect(B,aa(set(A),fun(B,bool),aTP_Lamp_ou(fun(B,set(A)),fun(set(A),fun(B,bool)),F2),X7)))) ) ).

% UN_image_subset
tff(fact_5917_INF__filter__not__bot,axiom,
    ! [I6: $tType,A: $tType,B4: set(I6),F3: fun(I6,filter(A))] :
      ( ! [X8: set(I6)] :
          ( pp(aa(set(I6),bool,aa(set(I6),fun(set(I6),bool),ord_less_eq(set(I6)),X8),B4))
         => ( pp(aa(set(I6),bool,finite_finite(I6),X8))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image(I6,filter(A),F3),X8)) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image(I6,filter(A),F3),B4)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_5918_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S: 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)),S))
     => ( pp(aa(set(A),bool,finite_finite(A),S))
       => ( ? [N8: nat] :
              ( ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N8))
                 => ! [M4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N8))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N2))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F2,M4)),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),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5919_top__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( top(C)
     => ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).

% top_apply
tff(fact_5920_UNIV__I,axiom,
    ! [A: $tType,X: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),top_top(set(A)))) ).

% UNIV_I
tff(fact_5921_Int__UNIV,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = top_top(set(A)) )
    <=> ( ( A3 = top_top(set(A)) )
        & ( B4 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_5922_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_5923_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_5924_Pow__UNIV,axiom,
    ! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_5925_Collect__const,axiom,
    ! [A: $tType,P: bool] :
      ( ( pp(P)
       => ( collect(A,aTP_Lamp_ov(bool,fun(A,bool),P)) = top_top(set(A)) ) )
      & ( ~ pp(P)
       => ( collect(A,aTP_Lamp_ov(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_5926_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_diff
tff(fact_5927_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_5928_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( ( aa(set(A),set(A),image(A,A,F2),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5929_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_on(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_5930_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image(B,A,aTP_Lamp_oa(A,fun(B,A),X)),top_top(set(B))) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ).

% range_constant
tff(fact_5931_INT__simps_I3_J,axiom,
    ! [E4: $tType,F: $tType,C3: set(E4),A3: fun(E4,set(F)),B4: set(F)] :
      ( ( ( C3 = bot_bot(set(E4)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E4),set(set(F)),image(E4,set(F),aa(set(F),fun(E4,set(F)),aTP_Lamp_ow(fun(E4,set(F)),fun(set(F),fun(E4,set(F))),A3),B4)),C3)) = top_top(set(F)) ) )
      & ( ( C3 != bot_bot(set(E4)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E4),set(set(F)),image(E4,set(F),aa(set(F),fun(E4,set(F)),aTP_Lamp_ow(fun(E4,set(F)),fun(set(F),fun(E4,set(F))),A3),B4)),C3)) = 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)),aa(set(E4),set(set(F)),image(E4,set(F),A3),C3))),B4) ) ) ) ).

% INT_simps(3)
tff(fact_5932_INT__simps_I4_J,axiom,
    ! [G3: $tType,H4: $tType,C3: set(H4),A3: set(G3),B4: fun(H4,set(G3))] :
      ( ( ( C3 = bot_bot(set(H4)) )
       => ( aa(set(set(G3)),set(G3),complete_Inf_Inf(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),aa(fun(H4,set(G3)),fun(H4,set(G3)),aTP_Lamp_or(set(G3),fun(fun(H4,set(G3)),fun(H4,set(G3))),A3),B4)),C3)) = top_top(set(G3)) ) )
      & ( ( C3 != bot_bot(set(H4)) )
       => ( aa(set(set(G3)),set(G3),complete_Inf_Inf(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),aa(fun(H4,set(G3)),fun(H4,set(G3)),aTP_Lamp_or(set(G3),fun(fun(H4,set(G3)),fun(H4,set(G3))),A3),B4)),C3)) = aa(set(G3),set(G3),aa(set(G3),fun(set(G3),set(G3)),minus_minus(set(G3)),A3),aa(set(set(G3)),set(G3),complete_Sup_Sup(set(G3)),aa(set(H4),set(set(G3)),image(H4,set(G3),B4),C3))) ) ) ) ).

% INT_simps(4)
tff(fact_5933_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : sums(A,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_ox(fun(nat,A),fun(nat,A),F2)),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_5934_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))))
     => ~ ! [X3: B] : B2 != aa(B,A,F2,X3) ) ).

% rangeE
tff(fact_5935_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_oy(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F2),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_5936_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image(B,A,F2),top_top(set(B))))) ).

% rangeI
tff(fact_5937_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F2,X) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B))))) ) ).

% range_eqI
tff(fact_5938_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F3: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
         => ! [J2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),I5))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F3,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,I3)),aa(A,filter(B),F3,J2)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F3),I5)) = bot_bot(filter(B)) )
      <=> ? [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),I5))
            & ( aa(A,filter(B),F3,X5) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_5939_Int__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),top_top(set(A))) = A3 ).

% Int_UNIV_right
tff(fact_5940_Int__UNIV__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B4) = B4 ).

% Int_UNIV_left
tff(fact_5941_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_5942_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_oz(A,bool)) ).

% UNIV_def
tff(fact_5943_UNIV__eq__I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
     => ( top_top(set(A)) = A3 ) ) ).

% UNIV_eq_I
tff(fact_5944_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X3: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),top_top(set(A)))) ).

% UNIV_witness
tff(fact_5945_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A2)) ) ).

% top.extremum_strict
tff(fact_5946_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_5947_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),top_top(set(A)))) ).

% subset_UNIV
tff(fact_5948_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),top_top(A))) ) ).

% top_greatest
tff(fact_5949_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5950_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5951_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),insert(A,X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_5952_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_5953_Inf__filter__not__bot,axiom,
    ! [A: $tType,B4: set(filter(A))] :
      ( ! [X8: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X8),B4))
         => ( pp(aa(set(filter(A)),bool,finite_finite(filter(A)),X8))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X8) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_5954_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B4: set(A),I: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))),B4))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,I)),B4)) ) ).

% range_subsetD
tff(fact_5955_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_5956_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_ord_atMost(A,H))) ) ).

% not_UNIV_le_Iic
tff(fact_5957_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_5958_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_5959_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_5960_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_5961_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: A,X: B] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) )
     => ( aa(B,A,F2,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_5962_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A3)))) ) ).

% surj_Compl_image_subset
tff(fact_5963_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aTP_Lamp_pa(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_5964_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_nb(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_5965_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_5966_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_5967_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_mq(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,P)))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,Q))) ) ) ).

% conj_subset_def
tff(fact_5968_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C3: 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)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C3))
     => 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)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),C3)) ) ).

% UN_finite_subset
tff(fact_5969_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5970_INT__extend__simps_I3_J,axiom,
    ! [F: $tType,E4: $tType,C3: set(E4),A3: fun(E4,set(F)),B4: set(F)] :
      ( ( ( C3 = bot_bot(set(E4)) )
       => ( 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)),aa(set(E4),set(set(F)),image(E4,set(F),A3),C3))),B4) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),top_top(set(F))),B4) ) )
      & ( ( C3 != bot_bot(set(E4)) )
       => ( 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)),aa(set(E4),set(set(F)),image(E4,set(F),A3),C3))),B4) = aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E4),set(set(F)),image(E4,set(F),aa(set(F),fun(E4,set(F)),aTP_Lamp_ow(fun(E4,set(F)),fun(set(F),fun(E4,set(F))),A3),B4)),C3)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_5971_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),aa(set(B),set(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),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_5972_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: 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)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),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)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_5973_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_ox(fun(nat,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_5974_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pb(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_5975_suminf__eq__SUP__real,axiom,
    ! [X7: fun(nat,real)] :
      ( summable(real,X7)
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,X7,I3)))
       => ( suminf(real,X7) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_pc(fun(nat,real),fun(nat,real),X7)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_5976_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_5977_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),finite_Fpow(B,A3))),finite_Fpow(A,B4))) ) ).

% image_Fpow_mono
tff(fact_5978_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_5979_card__UNIV__bool,axiom,
    aa(set(bool),nat,finite_card(bool),top_top(set(bool))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% card_UNIV_bool
tff(fact_5980_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))) ) )
      & ( ( A2 != zero_zero(real) )
       => ( aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = top_top(set(real)) ) ) ) ).

% range_mult
tff(fact_5981_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,top_top(fun(A,bool))) ).

% top_set_def
tff(fact_5982_infinite__UNIV__int,axiom,
    ~ pp(aa(set(int),bool,finite_finite(int),top_top(set(int)))) ).

% infinite_UNIV_int
tff(fact_5983_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less(filter(A)),F3),F9))
    <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9))
        & ~ pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F9),F3)) ) ) ).

% less_filter_def
tff(fact_5984_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_5985_surj__prod__encode,axiom,
    aa(set(product_prod(nat,nat)),set(nat),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_5986_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_5987_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(set(int),set(int),image(int,int,abs_abs(int)),top_top(set(int))))) ).

% int_in_range_abs
tff(fact_5988_Fpow__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),finite_Fpow(A,B4))) ) ).

% Fpow_mono
tff(fact_5989_Fpow__subset__Pow,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),pow2(A,A3))) ).

% Fpow_subset_Pow
tff(fact_5990_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = collect(set(A),aTP_Lamp_pd(set(A),fun(set(A),bool),A3)) ).

% Fpow_def
tff(fact_5991_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_pe(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_5992_card__UNIV__char,axiom,
    aa(set(char),nat,finite_card(char),top_top(set(char))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))) ).

% card_UNIV_char
tff(fact_5993_top1I,axiom,
    ! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).

% top1I
tff(fact_5994_UNIV__bool,axiom,
    top_top(set(bool)) = aa(set(bool),set(bool),insert(bool,fFalse),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))) ).

% UNIV_bool
tff(fact_5995_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_5996_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),modulo_modulo(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),N) ) ).

% char_of_mod_256
tff(fact_5997_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_5998_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,N: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),N) )
        <=> ( modulo_modulo(A,M,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = modulo_modulo(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_5999_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),N))
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_6000_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A2)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ).

% of_char_of
tff(fact_6001_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_6002_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_6003_char_Osize_I2_J,axiom,
    ! [X15: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : aa(char,nat,size_size(char),char2(X15,X2,X33,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size(2)
tff(fact_6004_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_6005_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% range_nat_of_char
tff(fact_6006_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),N) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),N) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_6007_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B22))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B1))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).

% integer_of_char_code
tff(fact_6008_char_Osize__gen,axiom,
    ! [X15: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_char(char2(X15,X2,X33,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size_gen
tff(fact_6009_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_6010_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: 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))
             => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,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)))
               => ( D4 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,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))
               => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6011_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,S2: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),aa(set(A),set(A),image(A,A,G),S2)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_image_chain
tff(fact_6012_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y4: A,X: A,S2: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,Y4,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
           => has_field_derivative(A,F2,Y4,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_field_derivative_subset
tff(fact_6013_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F8: A,X: A,S2: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,F8,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
           => has_field_derivative(A,F2,F8,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% DERIV_subset
tff(fact_6014_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T2: set(A),X: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T2))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),topolo174197925503356063within(A,X,S2)),topolo174197925503356063within(A,X,T2))) ) ) ).

% at_le
tff(fact_6015_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => ( ( 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))),D4)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_inverse'
tff(fact_6016_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pg(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),D4),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% DERIV_divide
tff(fact_6017_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ph(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),D4),E5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_diff
tff(fact_6018_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F8: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F8,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ph(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F8),G4),F3) ) ) ) ).

% field_differentiable_diff
tff(fact_6019_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_pi(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_6020_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pj(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D4),C2),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_cdivide
tff(fact_6021_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,X: A,S2: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,X,S2)) ) ).

% DERIV_cmult_Id
tff(fact_6022_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => 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),times_times(A),D4),C2),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_cmult_right
tff(fact_6023_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pl(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_cmult
tff(fact_6024_has__field__derivative__sinh,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [G: fun(A10,A10),Db: A10,X: A10,S2: set(A10)] :
          ( has_field_derivative(A10,G,Db,topolo174197925503356063within(A10,X,S2))
         => has_field_derivative(A10,aTP_Lamp_pm(fun(A10,A10),fun(A10,A10),G),aa(A10,A10,aa(A10,fun(A10,A10),times_times(A10),cosh(A10,aa(A10,A10,G,X))),Db),topolo174197925503356063within(A10,X,S2)) ) ) ).

% has_field_derivative_sinh
tff(fact_6025_has__field__derivative__cosh,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [G: fun(A10,A10),Db: A10,X: A10,S2: set(A10)] :
          ( has_field_derivative(A10,G,Db,topolo174197925503356063within(A10,X,S2))
         => has_field_derivative(A10,aTP_Lamp_pn(fun(A10,A10),fun(A10,A10),G),aa(A10,A10,aa(A10,fun(A10,A10),times_times(A10),sinh(A10,aa(A10,A10,G,X))),Db),topolo174197925503356063within(A10,X,S2)) ) ) ).

% has_field_derivative_cosh
tff(fact_6026_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S2))
           => 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),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_mult'
tff(fact_6027_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,X: A,S2: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S2))
           => 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),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_mult
tff(fact_6028_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6029_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                   => 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),plus_plus(real),X),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6030_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H5)),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                   => 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),H5))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6031_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H5)),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                   => 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),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6032_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G4: fun(real,real)] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
         => has_field_derivative(real,G,aa(real,real,G4,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,G4,X3))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,G,A2)),aa(real,real,G,B2))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6033_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3)) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6034_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),zero_zero(real))) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6035_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6036_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6037_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,aa(A,A,F2,X),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pp(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D4),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain'
tff(fact_6038_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain2
tff(fact_6039_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F8: A,X: A] :
          ( ! [X3: A] : has_field_derivative(A,G,aa(A,A,G4,X3),topolo174197925503356063within(A,X3,top_top(set(A))))
         => ( has_field_derivative(A,F2,F8,topolo174197925503356063within(A,X,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pq(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F8),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6040_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S2: set(A),G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F8: A,X: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
             => has_field_derivative(A,G,aa(A,A,G4,X3),topolo174197925503356063within(A,X3,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F8,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,F2,X)),S2))
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pq(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F8),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6041_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pr(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6042_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_ps(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6043_DERIV__isconst__all,axiom,
    ! [F2: fun(real,real),X: real,Y4: real] :
      ( ! [X3: real] : has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( aa(real,real,F2,X) = aa(real,real,F2,Y4) ) ) ).

% DERIV_isconst_all
tff(fact_6044_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                 => 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),plus_plus(real),X),H5)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6045_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6046_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_6047_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_6048_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))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                 => 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),H5))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6049_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)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H5),D3))
                 => 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),H5)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6050_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y4: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),set_or5935395276787703475ssThan(real,A2,B2)))
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y4),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( ! [X3: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F2,X) = aa(real,real,F2,Y4) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_6051_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain
tff(fact_6052_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F8: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => has_field_derivative(real,F2,aa(real,real,F8,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F8,Z2)) ) ) ) ) ).

% MVT2
tff(fact_6053_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))
       => ( ! [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))),D2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6054_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_6055_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pt(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,X)))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6056_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),N: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_pu(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),N))),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_power_Suc
tff(fact_6057_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,V,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,A2)),aa(real,real,V,B2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ) ).

% DERIV_const_average
tff(fact_6058_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S2: 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,S2)) ) ) ).

% DERIV_inverse
tff(fact_6059_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S2: set(A),N: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_pv(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_power
tff(fact_6060_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))
       => ( ! [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))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,X)),aa(real,real,F2,Y))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6061_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))
       => ( ! [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))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,Y)),aa(real,real,F2,X))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6062_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_6063_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set(real)] : has_field_derivative(real,aTP_Lamp_pw(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,S2)) ).

% DERIV_pow
tff(fact_6064_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),Y))
         => has_field_derivative(A,aTP_Lamp_px(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_6065_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,X: real,N: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_py(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))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_6066_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S2: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pg(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,S2)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6067_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S2))
         => ( ( 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,S2)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6068_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),F2: fun(A,A),F8: A,Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),Z2),aa(A,A,F2,Z2)) )
         => ( has_field_derivative(A,F2,F8,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F8) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6069_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => has_field_derivative(real,aTP_Lamp_pz(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6070_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F8: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L6: fun(nat,real)] :
      ( ! [N2: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_qa(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N2),aa(nat,real,aa(real,fun(nat,real),F8,X0),N2),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
           => summable(real,aa(real,fun(nat,real),F2,X3)) )
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( summable(real,aa(real,fun(nat,real),F8,X0))
           => ( summable(real,L6)
             => ( ! [N2: nat,X3: real,Y: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
                   => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y),set_or5935395276787703475ssThan(real,A2,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,X3),N2)),aa(nat,real,aa(real,fun(nat,real),F2,Y),N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L6,N2)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X3),Y))))) ) )
               => has_field_derivative(real,aTP_Lamp_qb(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F8,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6071_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
           => has_field_derivative(A,aTP_Lamp_px(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6072_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => has_field_derivative(A,aTP_Lamp_px(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6073_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qc(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
               => has_field_derivative(A,aTP_Lamp_px(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6074_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,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6075_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,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_qd(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,X),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6076_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_qe(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G,X)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,X))),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6077_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_6078_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_6079_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_6080_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_6081_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_6082_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_6083_has__field__derivative__tanh,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [G: fun(A10,A10),X: A10,Db: A10,S2: set(A10)] :
          ( ( cosh(A10,aa(A10,A10,G,X)) != zero_zero(A10) )
         => ( has_field_derivative(A10,G,Db,topolo174197925503356063within(A10,X,S2))
           => has_field_derivative(A10,aTP_Lamp_qf(fun(A10,A10),fun(A10,A10),G),aa(A10,A10,aa(A10,fun(A10,A10),times_times(A10),aa(A10,A10,aa(A10,fun(A10,A10),minus_minus(A10),one_one(A10)),aa(nat,A10,aa(A10,fun(nat,A10),power_power(A10),aa(A10,A10,tanh(A10),aa(A10,A10,G,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A10,X,S2)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6084_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero(real) )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
           => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6085_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6086_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,real),fun(real,fun(nat,real)),F2),X3)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => has_field_derivative(real,aTP_Lamp_qi(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_qg(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6087_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_6088_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_6089_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_6090_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 )
        & ! [M4: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),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_qj(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),T4)),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_6091_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 )
     => ( ! [M4: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),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_qj(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_6092_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_6093_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 )
         => ( ! [M4: nat,T4: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),H),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),zero_zero(real)))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qk(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_6094_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 )
       => ( ! [M4: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ? [T4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H))
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qk(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_6095_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 )
         => ( ! [M4: nat,T4: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),H))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qk(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_6096_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) )
         => ( ! [M4: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4)))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T4)),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_qj(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_6097_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 )
     => ( ! [M4: nat,T4: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),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_qj(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),T4)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_6098_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ? [T4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),C2))
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,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_ql(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T4)),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),A2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6099_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
             => ? [T4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),B2))
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,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_ql(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),T4)),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_6100_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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),A2),X))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
                 => ( ( X != C2 )
                   => ? [T4: 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),T4))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),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),T4))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),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_qm(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),T4)),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_6101_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B4: real] :
      ( ! [M4: nat,T4: real] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M2: nat,T8: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T8))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T8),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_qo(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B4),M2),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T8)),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_qp(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T8)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T8),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T8,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6102_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_qq(real,real),suminf(real,aTP_Lamp_qr(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6103_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_6104_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S2: 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,G4,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aTP_Lamp_qs(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_qt(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6105_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S2: 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,G4,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aTP_Lamp_qu(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_qv(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6106_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S2: set(A),T2: set(A)] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
           => has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_derivative_subset
tff(fact_6107_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_6108_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: fun(A,A),F3: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D4,F3)
         => ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X3),D7) = aa(A,A,D4,X3)
           => has_field_derivative(A,F2,D7,F3) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_6109_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_def
tff(fact_6110_has__derivative__mult__left,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G: fun(C,A),G4: fun(C,A),F3: filter(C),Y4: A] :
          ( has_derivative(C,A,G,G4,F3)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_qw(fun(C,A),fun(A,fun(C,A)),G),Y4),aa(A,fun(C,A),aTP_Lamp_qw(fun(C,A),fun(A,fun(C,A)),G4),Y4),F3) ) ) ).

% has_derivative_mult_left
tff(fact_6111_has__derivative__mult__right,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G: fun(C,A),G4: fun(C,A),F3: filter(C),X: A] :
          ( has_derivative(C,A,G,G4,F3)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_qx(fun(C,A),fun(A,fun(C,A)),G),X),aa(A,fun(C,A),aTP_Lamp_qx(fun(C,A),fun(A,fun(C,A)),G4),X),F3) ) ) ).

% has_derivative_mult_right
tff(fact_6112_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_qy(B,fun(A,B),C2),aTP_Lamp_qz(A,B),F3) ) ).

% has_derivative_const
tff(fact_6113_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F8,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ra(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_ra(fun(A,B),fun(fun(A,B),fun(A,B)),F8),G4),F3) ) ) ) ).

% has_derivative_diff
tff(fact_6114_has__derivative__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [F2: fun(D,A),F8: fun(D,A),X: D,S2: set(D),G: fun(D,A),G4: fun(D,A)] :
          ( has_derivative(D,A,F2,F8,topolo174197925503356063within(D,X,S2))
         => ( has_derivative(D,A,G,G4,topolo174197925503356063within(D,X,S2))
           => has_derivative(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_rb(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G),aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_rc(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))))),F2),F8),X),G),G4),topolo174197925503356063within(D,X,S2)) ) ) ) ).

% has_derivative_mult
tff(fact_6115_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_qz(A,B),F3,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X4: A] : aa(A,B,F3,X4) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6116_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T2: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),S2: set(C),X: C,F8: fun(C,A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T2))
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X3),topolo174197925503356063within(A,X3,T2)) )
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S2)),T2))
           => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X),S2))
             => ( has_derivative(C,A,F2,F8,topolo174197925503356063within(C,X,S2))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_rd(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_re(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),X),F8),topolo174197925503356063within(C,X,S2)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_6117_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_rf(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rg(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_exp
tff(fact_6118_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_rh(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_sin
tff(fact_6119_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S2: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S2))
         => has_derivative(A,A,aTP_Lamp_rj(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_cosh
tff(fact_6120_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S2: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S2))
         => has_derivative(A,A,aTP_Lamp_rk(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_sinh
tff(fact_6121_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),F8: fun(C,A),X: C,S: set(C),G: fun(C,A),G4: fun(C,A)] :
          ( has_derivative(C,A,F2,F8,topolo174197925503356063within(C,X,S))
         => ( has_derivative(C,A,G,G4,topolo174197925503356063within(C,X,S))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_rl(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_rm(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F8),X),G),G4),topolo174197925503356063within(C,X,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6122_has__derivative__inverse,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),X: C,F8: fun(C,A),S: set(C)] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F8,topolo174197925503356063within(C,X,S))
           => has_derivative(C,A,aTP_Lamp_rn(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_ro(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),F2),X),F8),topolo174197925503356063within(C,X,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_6123_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_rp(A,fun(A,A),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_inverse'
tff(fact_6124_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F8: real,G: fun(A,real),X: A,G4: fun(A,real),S2: set(A)] :
          ( has_field_derivative(real,F2,F8,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rq(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_rr(real,fun(fun(A,real),fun(A,real)),F8),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6125_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rt(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_cos
tff(fact_6126_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S: set(A),N: nat] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_ru(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_rv(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F8),X),N),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_power
tff(fact_6127_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S2: 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,G4,topolo174197925503356063within(A,X,S2))
           => 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),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_ln
tff(fact_6128_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),F8: fun(C,A),X: C,S: set(C),G: fun(C,A),G4: fun(C,A)] :
          ( has_derivative(C,A,F2,F8,topolo174197925503356063within(C,X,S))
         => ( has_derivative(C,A,G,G4,topolo174197925503356063within(C,X,S))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ry(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_rz(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F8),X),G),G4),topolo174197925503356063within(C,X,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6129_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)),F8: fun(I6,fun(A,B)),X: A,S: 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),F8,I3),topolo174197925503356063within(A,X,S)) )
         => has_derivative(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_sb(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_sd(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),I5),F2),F8),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_prod
tff(fact_6130_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,X7: set(A),F2: fun(A,real),F8: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,X7))
         => ( has_derivative(A,real,F2,F8,topolo174197925503356063within(A,X,X7))
           => ( 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),X7))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_se(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_sf(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),X),F2),F8),topolo174197925503356063within(A,X,X7)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6131_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S2: 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,G4,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aTP_Lamp_sg(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_sh(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6132_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_si(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_sj(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_arctan
tff(fact_6133_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S2: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aTP_Lamp_sk(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_sl(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_tan
tff(fact_6134_has__derivative__floor,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [G: fun(A,real),X: A,F2: fun(real,Aa),G4: fun(A,real),S2: set(A)] :
          ( topolo3448309680560233919inuous(real,Aa,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))),F2)
         => ( ~ pp(aa(set(Aa),bool,aa(Aa,fun(set(Aa),bool),member(Aa),aa(real,Aa,F2,aa(A,real,G,X))),ring_1_Ints(Aa)))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_sm(fun(A,real),fun(fun(real,Aa),fun(A,real)),G),F2),aTP_Lamp_sn(fun(A,real),fun(A,real),G4),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6135_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qc(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sp(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_6136_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sq(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6137_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sr(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6138_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F3: 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_ss(nat,fun(fun(A,real),fun(A,real)),N),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_6139_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),X: A,S: set(A),T6: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),S))
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,X,T6)) ) ) ) ).

% tendsto_within_subset
tff(fact_6140_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_st(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivativeD
tff(fact_6141_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_st(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivative_iff
tff(fact_6142_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X3))) )
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,G,X3)),aa(A,real,F2,X3))) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6143_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_su(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_6144_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6145_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6146_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L6: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6147_LIM__not__zero,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & zero(Aa)
        & topological_t2_space(Aa) )
     => ! [K: Aa,A2: A] :
          ( ( K != zero_zero(Aa) )
         => ~ filterlim(A,Aa,aTP_Lamp_sw(Aa,fun(A,Aa),K),topolo7230453075368039082e_nhds(Aa,zero_zero(Aa)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_not_zero
tff(fact_6148_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sx(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6149_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & 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),A2))),D6)) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6150_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,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,X3)),M))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X3)),L)))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6151_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A),F24: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F23),F24))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F13),F12))
         => filterlim(A,B,F2,F24,F13) ) ) ) ).

% filterlim_mono
tff(fact_6152_tendsto__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: filter(B),F9: filter(B),F2: fun(B,A),L: A] :
          ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F3),F9))
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F9)
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mono
tff(fact_6153_tendsto__null__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(B),F2: fun(A,fun(B,C)),F3: filter(A)] :
          ( ! [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_sz(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_ta(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_6154_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tb(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_6155_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tb(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_6156_Lim__transform,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(B,A),A2: A,F3: filter(B),F2: fun(B,A)] :
          ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tc(fun(B,A),fun(fun(B,A),fun(B,A)),G),F2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform
tff(fact_6157_Lim__transform2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_td(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform2
tff(fact_6158_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tb(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_6159_Lim__transform__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),G: fun(B,A),F3: filter(B),A2: A] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_td(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
          <=> filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_6160_tendsto__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_te(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),A2),B2)),F3) ) ) ) ).

% tendsto_diff
tff(fact_6161_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_6162_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_6163_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_6164_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_6165_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( ( A2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_kn(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A2)),F3) ) ) ).

% tendsto_ln
tff(fact_6166_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_th(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_6167_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_th(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_6168_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_th(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_6169_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ti(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr
tff(fact_6170_tendsto__sgn,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( ( L != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_tj(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,sgn_sgn(A),L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_6171_tendsto__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( A2 != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_tk(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,inverse_inverse(A),A2)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_6172_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_tl(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F3) ) ) ) ).

% tendsto_tan
tff(fact_6173_tendsto__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cosh(A,A2) != zero_zero(A) )
           => filterlim(C,A,aTP_Lamp_tm(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tanh(A),A2)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_6174_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_tn(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F3) ) ) ) ).

% tendsto_cot
tff(fact_6175_tendsto__add__zero,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(D,B),F3: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_to(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_6176_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,X)),F3) ) ).

% tendsto_real_sqrt
tff(fact_6177_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A),N: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_tq(fun(A,real),fun(nat,fun(A,real)),F2),N),topolo7230453075368039082e_nhds(real,aa(real,real,root(N),X)),F3) ) ).

% tendsto_real_root
tff(fact_6178_continuous__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F3: filter(B),F2: fun(B,A),C2: A] :
          ( topolo3448309680560233919inuous(B,A,F3,F2)
         => topolo3448309680560233919inuous(B,A,F3,aa(A,fun(B,A),aTP_Lamp_tr(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_6179_continuous__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F3: filter(B),F2: fun(B,A),C2: A] :
          ( topolo3448309680560233919inuous(B,A,F3,F2)
         => topolo3448309680560233919inuous(B,A,F3,aa(A,fun(B,A),aTP_Lamp_ts(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_6180_tendsto__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_tt(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_6181_tendsto__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_tu(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_6182_continuous__mult_H,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(D),F2: fun(D,B),G: fun(D,B)] :
          ( topolo3448309680560233919inuous(D,B,F3,F2)
         => ( topolo3448309680560233919inuous(D,B,F3,G)
           => topolo3448309680560233919inuous(D,B,F3,aa(fun(D,B),fun(D,B),aTP_Lamp_tv(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_6183_continuous__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [F3: filter(D),F2: fun(D,A),G: fun(D,A)] :
          ( topolo3448309680560233919inuous(D,A,F3,F2)
         => ( topolo3448309680560233919inuous(D,A,F3,G)
           => topolo3448309680560233919inuous(D,A,F3,aa(fun(D,A),fun(D,A),aTP_Lamp_tw(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_6184_tendsto__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tx(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),F3) ) ) ) ).

% tendsto_mult
tff(fact_6185_tendsto__mult__one,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(D,B),F3: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_ty(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_6186_tendsto__mult__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F3: filter(D),G: fun(D,A)] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => ( filterlim(D,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_tz(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_6187_tendsto__mult__left__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F3: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_ua(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_6188_tendsto__mult__right__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F3: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_ub(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_6189_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uc(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_6190_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_ud(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_6191_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),N: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ue(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_6192_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
         => ( ( A2 != 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_uf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_6193_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,F2: fun(A,D),L6: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
          <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ug(A,fun(fun(A,D),fun(A,D)),A2),F2),topolo7230453075368039082e_nhds(D,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6194_IVT,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y4: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),Y4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y4),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F2) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F2,X3) = Y4 ) ) ) ) ) ) ) ).

% IVT
tff(fact_6195_IVT2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y4: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,B2)),Y4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y4),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F2) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F2,X3) = Y4 ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6196_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => ( 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),A2))),R))
                 => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6197_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & 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),X5),A2))),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,X5)),L6))),R5)) ) ) ) ) ) ).

% LIM_eq
tff(fact_6198_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L6: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & 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),A2))),S7)) )
                     => 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)),L6))),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6199_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [S3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S3))
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & 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),A2))),S3)) )
                   => 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)),L6))),R2)) ) ) ) ) ) ).

% LIM_D
tff(fact_6200_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D4: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uh(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ui(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6201_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ? [L7: real,M8: real] :
            ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L7),aa(real,real,F2,X4)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,X4)),M8)) ) )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L7),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),M8)) )
               => ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,real,F2,X3) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_6202_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X4)),zero_zero(real))) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6203_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => ( aa(real,real,F2,X4) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6204_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))
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F2,X4))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6205_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & 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),A2))),D6)) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6206_isCont__real__sqrt,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_6207_isCont__real__root,axiom,
    ! [X: real,N: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),root(N)) ).

% isCont_real_root
tff(fact_6208_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),aa(fun(A,B),fun(A,B),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6209_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_uk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_6210_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_ul(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_6211_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S2: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),aTP_Lamp_um(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_6212_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S2: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),aTP_Lamp_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_6213_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uo(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6214_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uo(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6215_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_up(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6216_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K5: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3)))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6217_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),aa(real,A,F2,X4))) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F2,X3) = M8 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6218_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X4)),M8)) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F2,X3) = M8 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6219_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M8: A] :
              ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X4)),M8)) ) ) ) ) ).

% isCont_bounded
tff(fact_6220_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
               => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_6221_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uo(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6222_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_6223_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6224_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_6225_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uq(fun(A,B),fun(A,fun(A,B)),F2),A2),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6226_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S2: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),aTP_Lamp_tl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_6227_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S2: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),aTP_Lamp_tn(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_6228_continuous__at__within__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: C,A3: set(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),F2)
         => ( ( cosh(A,aa(C,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),aTP_Lamp_ur(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_6229_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))))
        <=> ? [G5: 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,G5,Z5)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z5),X))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),G5)
              & ( aa(A,A,G5,X) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6230_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X4)),M8)) )
                & ! [N8: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),M8))
                   => ? [X3: real] :
                        ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),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(A),N8),aa(real,A,F2,X3))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6231_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_6232_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A,D2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F3,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6233_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F3: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F3,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6234_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_6235_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_6236_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S2))
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S2))
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F2,X3)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6237_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S2))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S2))
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F2,X3)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6238_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)
           => ( ! [H3: A,N2: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),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,H3),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N2)),real_V7770717601297561774m_norm(A,H3)))) ) )
             => filterlim(A,B,aTP_Lamp_us(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_6239_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_tl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_6240_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_ut(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_6241_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_tn(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_6242_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D4: real,G: fun(real,real),X: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D4,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D4 != zero_zero(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
           => ( ! [Y: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),B2))
                   => ( aa(real,real,F2,aa(real,real,G,Y)) = Y ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D4),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_6243_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_uu(fun(nat,A),fun(nat,fun(A,A)),C2),N)) ) ).

% isCont_polynom
tff(fact_6244_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),Y))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_px(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6245_LIM__less__bound,axiom,
    ! [B2: real,X: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),B2),X))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,B2,X)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X3))) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X))) ) ) ) ).

% LIM_less_bound
tff(fact_6246_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))
     => ( ! [Z2: 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),Z2),X))),D2))
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: 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),Z2),X))),D2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_6247_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F8: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [Z2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                 => has_field_derivative(real,G,aa(real,real,G4,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                   => has_field_derivative(real,F2,aa(real,real,F8,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G4,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F8,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6248_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [X: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(real,A,F2,X)),ring_1_Ints(A)))
           => has_field_derivative(real,aTP_Lamp_uv(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_6249_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_px(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6250_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,Aa),C2: fun(nat,Aa),K5: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_uw(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C2),K5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(Aa,aa(A,Aa,F2,A2))),real_V7770717601297561774m_norm(Aa,K5)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_uy(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6251_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real)))
         => ! [N7: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6252_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat))))
         => ! [N7: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6253_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_va(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_6254_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vb(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_6255_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vc(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_6256_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_vd(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_6257_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),N: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_ve(fun(A,real),fun(nat,fun(A,real)),F2),N)) ) ) ).

% continuous_real_root
tff(fact_6258_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6259_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_vf(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6260_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_vg(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_6261_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_vg(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6262_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_vh(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_6263_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X7: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6264_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X7: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(nat,A,X7,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6265_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N5: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(nat,A,F2,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),L)) ) ) ) ).

% Lim_bounded2
tff(fact_6266_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M10: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),C3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),C3)) ) ) ) ).

% Lim_bounded
tff(fact_6267_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X7: fun(nat,A),X: A,Y6: fun(nat,A),Y4: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y4),at_top(nat))
           => ( ? [N8: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),aa(nat,A,Y6,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6268_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N5: nat,X7: fun(nat,A),Y6: fun(nat,A),X: A,Y4: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),aa(nat,A,Y6,N2))) )
         => ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y4),at_top(nat))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ) ).

% lim_mono
tff(fact_6269_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S2))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))) ) ) ) ).

% Sup_lim
tff(fact_6270_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S2))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)) ) ) ) ).

% Inf_lim
tff(fact_6271_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_6272_continuous__at__within__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A2: C,S2: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S2),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S2),G)
           => ( ( aa(C,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S2),aa(fun(C,real),fun(C,real),aTP_Lamp_vi(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_6273_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S2: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S2),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S2),aTP_Lamp_vj(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_6274_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_vk(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6275_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_6276_monoseq__convergent,axiom,
    ! [X7: fun(nat,real),B4: real] :
      ( topological_monoseq(real,X7)
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X7,I3))),B4))
       => ~ ! [L7: real] : ~ filterlim(nat,real,X7,topolo7230453075368039082e_nhds(real,L7),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_6277_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_vl(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6278_isCont__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A2: C,F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),G)
           => ( ( aa(C,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),aa(fun(C,real),fun(C,real),aTP_Lamp_vi(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_6279_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_vj(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_6280_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),X: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N7)),X))
                & ! [M2: nat,N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N7))) ) )
              | ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,A2,N7)))
                & ! [M2: nat,N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N7)),aa(nat,A,A2,M2))) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6281_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_vm(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6282_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vn(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_6283_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X7: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X7,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_vo(fun(nat,A),fun(nat,fun(nat,A)),X7),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6284_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_vp(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_6285_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_vq(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_6286_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_vr(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N7)),L2))
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L2),aa(nat,real,G,N7)))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6287_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R3),aa(nat,real,X7,N2))) )
     => filterlim(nat,real,aTP_Lamp_vs(fun(nat,real),fun(nat,real),X7),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6288_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_vt(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6289_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_vu(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6290_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_vv(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6291_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_vw(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6292_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S2: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),aa(fun(A,real),fun(A,real),aTP_Lamp_vx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6293_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))
             => ? [N7: 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,N7)),E))) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6294_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vy(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6295_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vz(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6296_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_wa(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6297_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_6298_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_vp(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_6299_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_vq(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_6300_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: 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_wb(real,fun(real,fun(nat,real)),X),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6301_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_6302_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_6303_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_wc(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6304_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S2: A] :
          ( sums(A,F2,S2)
        <=> filterlim(nat,A,aTP_Lamp_wd(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S2),at_top(nat)) ) ) ).

% sums_def'
tff(fact_6305_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_we(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6306_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_vx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6307_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No: nat] :
                ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),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,X7,N4)),L6))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6308_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L6: 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,X7,N2)),L6))),R3)) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6309_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L6: A,R2: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [No3: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N7))
               => 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,X7,N7)),L6))),R2)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6310_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_6311_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_wf(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6312_tendsto__at__iff__sequentially,axiom,
    ! [C: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,C),A2: C,X: A,S2: set(A)] :
          ( filterlim(A,C,F2,topolo7230453075368039082e_nhds(C,A2),topolo174197925503356063within(A,X,S2))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,X9,I2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,X),at_top(nat))
               => filterlim(nat,C,aa(fun(nat,A),fun(nat,C),comp(A,C,nat,F2),X9),topolo7230453075368039082e_nhds(C,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_6313_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(B,nat),F3: filter(B),X: A] :
          ( filterlim(B,nat,F2,at_top(nat),F3)
         => ( pp(aa(real,bool,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_wg(fun(B,nat),fun(A,fun(B,A)),F2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_6314_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_wh(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6315_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_6316_summable__Leibniz_I1_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => summable(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6317_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S2: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N2: nat] : aa(nat,A,S2,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_wi(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S2),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6318_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_wj(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6319_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_wk(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6320_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => summable(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6321_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_wl(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_wm(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_6322_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_wn(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_wm(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_6323_summable__Leibniz_I4_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_wo(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6324_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_6325_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,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_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6326_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_wo(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6327_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N7: 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_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)))),L2))
              & filterlim(nat,real,aTP_Lamp_wo(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N7: 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_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)),one_one(nat))))))
              & filterlim(nat,real,aTP_Lamp_wp(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6328_summable__Leibniz_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_wp(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6329_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_wp(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6330_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uz(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6331_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F8),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6332_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D4: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_wr(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D4),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6333_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_ra(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_6334_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_qz(A,B)) ) ).

% bounded_linear_zero
tff(fact_6335_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y4: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ws(A,fun(A,A),Y4)) ) ).

% bounded_linear_divide
tff(fact_6336_real__bounded__linear,axiom,
    ! [F2: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F2)
    <=> ? [C6: real] :
        ! [X5: real] : aa(real,real,F2,X5) = aa(real,real,aa(real,fun(real,real),times_times(real),X5),C6) ) ).

% real_bounded_linear
tff(fact_6337_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),X)) ) ).

% bounded_linear_mult_right
tff(fact_6338_bounded__linear__mult__const,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,A),Y4: A] :
          ( real_V3181309239436604168linear(C,A,G)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_qw(fun(C,A),fun(A,fun(C,A)),G),Y4)) ) ) ).

% bounded_linear_mult_const
tff(fact_6339_bounded__linear__const__mult,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,A),X: A] :
          ( real_V3181309239436604168linear(C,A,G)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_qx(fun(C,A),fun(A,fun(C,A)),G),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_6340_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y4: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_wt(A,fun(A,A),Y4)) ) ).

% bounded_linear_mult_left
tff(fact_6341_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K10: real] :
            ! [X4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K10))) ) ) ).

% bounded_linear.bounded
tff(fact_6342_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wu(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6343_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K10: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K10))
              & ! [X4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K10))) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6344_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K10: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K10))
              & ! [X4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K10))) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6345_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K5: real] :
          ( ! [X3: A,Y: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))
         => ( ! [R3: real,X3: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X3))
           => ( ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K5)))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6346_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S2: set(A)] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_wv(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F8),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6347_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S2: set(A)] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F8),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_at_within
tff(fact_6348_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F8: fun(A,B),X: A,F2: fun(A,B),S2: set(A)] :
          ( real_V3181309239436604168linear(A,B,F8)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_wx(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F8),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2))
           => has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivativeI
tff(fact_6349_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & ? [E3: fun(A,B)] :
                ( ! [H6: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F8,H6))),aa(A,B,E3,H6))
                & filterlim(A,real,aTP_Lamp_wy(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_6350_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S2: set(A)] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F8),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_within
tff(fact_6351_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F8,F3)
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_xa(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F8),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_6352_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S: set(A),F2: fun(A,B),F8: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
         => ( topolo1002775350975398744n_open(A,S)
           => ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S))
            <=> ( real_V3181309239436604168linear(A,B,F8)
                & ? [E3: fun(A,B)] :
                    ( ! [H6: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)),S))
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F8,H6))),aa(A,B,E3,H6)) ) )
                    & filterlim(A,real,aTP_Lamp_wy(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_6353_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
        ? [A7: fun(nat,set(A))] :
          ( ! [I4: nat] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I4)))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I4)) )
          & ! [S8: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S8)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S8)) )
             => ? [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),A7,I3)),S8)) ) ) ) ).

% first_countable_basis
tff(fact_6354_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),T9))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),S)) ) ) ) ) ).

% open_subopen
tff(fact_6355_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
             => ? [T10: set(A)] :
                  ( topolo1002775350975398744n_open(A,T10)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T10))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),S)) ) )
         => topolo1002775350975398744n_open(A,S) ) ) ).

% openI
tff(fact_6356_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S: set(A),T6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
         => ( topolo1002775350975398744n_open(A,S)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T6))
             => ( topolo174197925503356063within(A,A2,T6) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_6357_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y4: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [B3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B3))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,X,B3)),S)) ) ) ) ) ) ).

% open_right
tff(fact_6358_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y4: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [B3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),X))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B3,X)),S)) ) ) ) ) ) ).

% open_left
tff(fact_6359_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( topolo1002775350975398744n_open(A,S9)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),F0),S9))
               => ? [N6: nat] :
                  ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N4))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F2,N4)),S9)) ) ) ) ) ) ).

% lim_explicit
tff(fact_6360_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6361_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_um(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6362_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6363_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S: set(A),T6: set(A),U3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
         => ( topolo1002775350975398744n_open(A,S)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T6),S)),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)),U3),S)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,X,T6) = topolo174197925503356063within(A,X,U3) ) ) ) ) ) ).

% at_within_nhd
tff(fact_6364_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_xc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_6365_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_vj(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_6366_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xd(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_tl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_6367_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xd(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_tn(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_6368_continuous__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F3,F2)
         => ( ( cosh(A,aa(C,A,F2,topolo3827282254853284352ce_Lim(C,C,F3,aTP_Lamp_xe(C,C)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_ur(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_tanh
tff(fact_6369_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,S: set(A),F2: fun(A,D),L6: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
           => ( topolo1002775350975398744n_open(A,S)
             => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L6),topolo174197925503356063within(A,A2,S))
              <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ug(A,fun(fun(A,D),fun(A,D)),A2),F2),topolo7230453075368039082e_nhds(D,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6370_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A)))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(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,F3,aTP_Lamp_xb(A,A)))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_vx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6371_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F8: fun(A,B),S2: set(A),X: A,F2: fun(A,B),H7: fun(A,real)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => ( real_V3181309239436604168linear(A,B,F8)
           => ( ! [Y: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
                 => ( ( Y != X )
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,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,Y)),aa(A,B,F2,X))),aa(A,B,F8,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),X))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),X)))),aa(A,real,H7,Y))) ) ) )
             => ( filterlim(A,real,H7,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S2))
               => has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6372_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_6373_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] :
          ( ( real_V557655796197034286t_dist(A,X,Y4) = zero_zero(real) )
        <=> ( X = Y4 ) ) ) ).

% dist_eq_0_iff
tff(fact_6374_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,X,X) = zero_zero(real) ) ).

% dist_self
tff(fact_6375_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_6376_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y4)))
        <=> ( X != Y4 ) ) ) ).

% zero_less_dist_iff
tff(fact_6377_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y4)),zero_zero(real)))
        <=> ( X = Y4 ) ) ) ).

% dist_le_zero_iff
tff(fact_6378_dist__diff_I1_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V557655796197034286t_dist(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(1)
tff(fact_6379_dist__diff_I2_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),A2) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(2)
tff(fact_6380_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: real,A2: A,Y4: real] : real_V557655796197034286t_dist(A,aa(A,A,real_V8093663219630862766scaleR(A,X),A2),aa(A,A,real_V8093663219630862766scaleR(A,Y4),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4))),real_V7770717601297561774m_norm(A,A2)) ) ).

% dist_scaleR
tff(fact_6381_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y4))) ) ).

% zero_le_dist
tff(fact_6382_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Y4)),real_V557655796197034286t_dist(A,Y4,Z)))) ) ).

% dist_triangle
tff(fact_6383_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A,Z: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y4)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y4,Z)))) ) ).

% dist_triangle2
tff(fact_6384_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A,A2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y4)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,X)),real_V557655796197034286t_dist(A,A2,Y4)))) ) ).

% dist_triangle3
tff(fact_6385_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y4: 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_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y4,Z))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y4)),E2)) ) ) ).

% dist_triangle_le
tff(fact_6386_dist__norm,axiom,
    ! [A: $tType] :
      ( real_V6936659425649961206t_norm(A)
     => ! [X: A,Y4: A] : real_V557655796197034286t_dist(A,X,Y4) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y4)) ) ).

% dist_norm
tff(fact_6387_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_6388_dist__real__def,axiom,
    ! [X: real,Y4: real] : real_V557655796197034286t_dist(real,X,Y4) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y4)) ).

% dist_real_def
tff(fact_6389_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y4)),zero_zero(real))) ) ).

% dist_not_less_zero
tff(fact_6390_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y4: A] :
          ( ( X != Y4 )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y4))) ) ) ).

% dist_pos_lt
tff(fact_6391_dist__complex__def,axiom,
    ! [X: complex,Y4: complex] : real_V557655796197034286t_dist(complex,X,Y4) = real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),X),Y4)) ).

% dist_complex_def
tff(fact_6392_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
             => ? [E3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
                  & ! [Y5: A] :
                      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y5,X5)),E3))
                     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y5),S)) ) ) ) ) ) ).

% open_dist
tff(fact_6393_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: 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,A2,B2)),real_V557655796197034286t_dist(A,B2,C2)))),real_V557655796197034286t_dist(A,A2,C2))) ) ).

% abs_dist_diff_le
tff(fact_6394_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F8: A,A2: A,S: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F8,topolo174197925503356063within(A,A2,S))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D2))
                     => ( aa(A,A,F2,X3) = aa(A,A,G,X3) ) ) )
               => has_field_derivative(A,G,F8,topolo174197925503356063within(A,A2,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_6395_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F8: fun(A,B),X: A,S2: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F8,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
             => ( ! [X10: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X10),S2))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D2))
                     => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) )
               => has_derivative(A,B,G,F8,topolo174197925503356063within(A,X,S2)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_6396_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M4))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N2))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% metric_CauchyI
tff(fact_6397_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ? [M8: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M2))
               => ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N7))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M2),aa(nat,A,X7,N7))),E2)) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6398_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S2)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [N6: nat] :
                ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S2,N4),aa(nat,A,S2,N6))),E3)) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6399_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [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))
                 => ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N4))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M5),aa(nat,A,X7,N4))),E3)) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6400_dist__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M: int,N: int] : real_V557655796197034286t_dist(A,aa(int,A,ring_1_of_int(A),M),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),M),N))) ) ).

% dist_of_int
tff(fact_6401_metric__LIM__imp__LIM,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(C,A),L: A,A2: C,G: fun(C,B),M: B] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(C,A2,top_top(set(C))))
         => ( ! [X3: C] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(C,B,G,X3),M)),real_V557655796197034286t_dist(A,aa(C,A,F2,X3),L))) )
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(C,A2,top_top(set(C)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_6402_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,X: A,S: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( ! [X10: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X10),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X10,X)))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D2))
                     => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_6403_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y4: A,E2: real,X2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,Y4)),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,X2,Y4)),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,X2)),E2)) ) ) ) ).

% dist_triangle_half_l
tff(fact_6404_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y4: A,X15: A,E2: real,X2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y4,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,Y4,X2)),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,X2)),E2)) ) ) ) ).

% dist_triangle_half_r
tff(fact_6405_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X2: A,E2: real,X33: A,X42: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X2)),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,X2,X33)),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,X33,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_6406_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_6407_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G6: filter(B),X: A,S: set(A),F3: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G6,topolo174197925503356063within(A,X,S))
         => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),G6),F3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
             => ( ! [X10: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X10),S))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X10,X)))
                     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X10,X)),D2))
                       => ( aa(A,B,F2,X10) = aa(A,B,G,X10) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_6408_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))
                 => ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N4))
                     => 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,N4))),E3)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6409_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M9: nat] :
                ! [M4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M4))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N2))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI'
tff(fact_6410_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M: nat,N: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))) ) ).

% dist_of_nat
tff(fact_6411_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_lessThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_6412_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_xf(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_6413_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),R))
                   => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_6414_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L6: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S7)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L6)),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_6415_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [S3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S3))
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S3)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L6)),R2)) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_6416_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),S6)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X5),L6)),R5)) ) ) ) ) ) ).

% LIM_def
tff(fact_6417_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No: nat] :
                ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N4),L6)),R5)) ) ) ) ) ).

% lim_sequentially
tff(fact_6418_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L6: 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_V557655796197034286t_dist(A,aa(nat,A,X7,N2),L6)),R3)) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6419_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L6: A,R2: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [No3: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N7))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N7),L6)),R2)) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6420_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [J3: nat] :
            ? [M7: nat] :
            ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),M5))
             => ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M5),aa(nat,A,X7,N4))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6421_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D6)) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xg(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_6422_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6423_metric__isCont__LIM__compose2,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [A2: A,F2: fun(A,C),G: fun(C,D),L: D] :
          ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(C,D,G,topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(C,aa(A,C,F2,A2),top_top(set(C))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D6)) )
                     => ( aa(A,C,F2,X3) != aa(A,C,F2,A2) ) ) )
             => filterlim(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_xi(fun(A,C),fun(fun(C,D),fun(A,D)),F2),G),topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_6424_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),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))
                  & ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N4))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N4),L6)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6425_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6426_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: fun(real,real),F3: 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),F3)
       => ( ~ 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_xj(nat,fun(fun(real,real),fun(real,real)),N),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6427_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [K3: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),K3))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_xl(real,fun(A,set(A)),E3)),K3)))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_6428_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_xm(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_6429_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_greaterThan(A,X)),set_ord_greaterThan(A,Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_6430_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A),T6: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),S))
           => topolo6688025880775521714ounded(A,T6) ) ) ) ).

% totally_bounded_subset
tff(fact_6431_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_xn(fun(real,A),fun(real,fun(real,A)),F2),A2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_6432_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),P3: A,F12: filter(B),C2: A,L: A] :
          ( filterlim(B,A,F2,topolo174197925503356063within(A,P3,set_ord_greaterThan(A,P3)),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),P3) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_xo(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo174197925503356063within(A,L,set_ord_greaterThan(A,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6433_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_6434_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_6435_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_xp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_6436_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_6437_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,S2: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,X,S2) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_xq(A,fun(set(A),fun(A,filter(A))),X),S2)),set_ord_greaterThan(A,X)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_xr(A,fun(set(A),fun(A,filter(A))),X),S2)),set_ord_lessThan(A,X)))) ) ) ) ).

% at_within_order
tff(fact_6438_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A4: A,B3: A,X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),S))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),S))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S)) ) ) ) )
         => ? [A4: A,B3: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = set_ord_lessThan(A,B3) )
              | ( S = set_ord_atMost(A,B3) )
              | ( S = set_ord_greaterThan(A,A4) )
              | ( S = set_ord_atLeast(A,A4) )
              | ( S = set_or5935395276787703475ssThan(A,A4,B3) )
              | ( S = set_or3652927894154168847AtMost(A,A4,B3) )
              | ( S = set_or7035219750837199246ssThan(A,A4,B3) )
              | ( S = set_or1337092689740270186AtMost(A,A4,B3) ) ) ) ) ).

% interval_cases
tff(fact_6439_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_6440_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atLeast(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I)) ) ) ).

% atLeast_iff
tff(fact_6441_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,X)),set_ord_atLeast(A,Y4)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ).

% atLeast_subset_iff
tff(fact_6442_principal__le__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),principal(A,A3)),principal(A,B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ).

% principal_le_iff
tff(fact_6443_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L4)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_6444_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),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_6445_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_ord_atLeast(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_minus_const_atLeast
tff(fact_6446_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_6447_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_greaterThan(A,A2)),set_ord_atLeast(A,A2))) ) ).

% Ioi_le_Ico
tff(fact_6448_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L4: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,L)),set_or1337092689740270186AtMost(A,L4,H2))) ) ).

% not_Ici_le_Icc
tff(fact_6449_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = collect(A,aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_6450_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_ord_atLeast(A,L))) ) ).

% not_UNIV_le_Ici
tff(fact_6451_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L4: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,H)),set_ord_atLeast(A,L4))) ) ).

% not_Iic_le_Ici
tff(fact_6452_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,L)),set_ord_atMost(A,H2))) ) ).

% not_Ici_le_Iic
tff(fact_6453_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,A2)),set_ord_greaterThan(A,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6454_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_6455_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_6456_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_6457_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F3: fun(A,set(B)),F2: fun(B,C),G6: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
           => ! [J2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
               => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,I3)),aa(A,set(B),F3,J2)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,J2)),aa(A,set(B),F3,I3))) ) ) )
       => ( filterlim(B,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image(D,filter(C),aTP_Lamp_xs(fun(D,set(C)),fun(D,filter(C)),G6)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_xt(fun(A,set(B)),fun(A,filter(B)),F3)),I5)))
        <=> ! [X5: D] :
              ( pp(aa(set(D),bool,aa(D,fun(set(D),bool),member(D),X5),J4))
             => ? [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),I5))
                  & ! [Xb3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xb3),aa(A,set(B),F3,Xa4)))
                     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,F2,Xb3)),aa(D,set(C),G6,X5))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_6458_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A)] : topolo174197925503356063within(A,A2,S2) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_6459_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_xv(A,fun(real,filter(A)),X)),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% nhds_metric
tff(fact_6460_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S2: set(A)] : topolo174197925503356063within(A,X,S2) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_xw(A,fun(set(A),fun(set(A),filter(A))),X),S2)),collect(set(A),aTP_Lamp_xx(A,fun(set(A),bool),X)))) ) ).

% at_within_eq
tff(fact_6461_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: fun(real,real),F3: 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),F3)
       => ( 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_xj(nat,fun(fun(real,real),fun(real,real)),N),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6462_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_6463_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_6464_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_6465_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_6466_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F3: 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),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(nat,fun(fun(A,real),fun(A,real)),N),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_6467_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xy(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_6468_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_6469_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F3: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F3)
    <=> filterlim(A,real,aTP_Lamp_aa(fun(A,nat),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_6470_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6471_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ya(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6472_tendsto__neg__powr,axiom,
    ! [A: $tType,S2: real,F2: fun(A,real),F3: filter(A)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),S2),zero_zero(real)))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yb(real,fun(fun(A,real),fun(A,real)),S2),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_6473_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_yc(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_6474_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_yd(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_6475_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_yd(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_6476_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_6477_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_6478_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6479_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ye(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6480_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_yf(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6481_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),X3))
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6482_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_6483_at__infinity__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( at_infinity(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_yh(real,filter(A))),top_top(set(real)))) ) ) ).

% at_infinity_def
tff(fact_6484_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F8: fun(real,real),Y4: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_6485_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_yl(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_6486_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6487_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N6),N4))
             => pp(aa(A,bool,P,N4)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6488_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),X3))
             => pp(aa(A,bool,P,X3)) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6489_at__top__le__at__infinity,axiom,
    pp(aa(filter(real),bool,aa(filter(real),fun(filter(real),bool),ord_less_eq(filter(real)),at_top(real)),at_infinity(real))) ).

% at_top_le_at_infinity
tff(fact_6490_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B7: real] :
            ! [X5: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X5)))
             => pp(aa(A,bool,P,X5)) ) ) ) ).

% eventually_at_infinity
tff(fact_6491_filter__leD,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A),P: fun(A,bool)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9))
     => ( eventually(A,P,F9)
       => eventually(A,P,F3) ) ) ).

% filter_leD
tff(fact_6492_filter__leI,axiom,
    ! [A: $tType,F9: filter(A),F3: filter(A)] :
      ( ! [P5: fun(A,bool)] :
          ( eventually(A,P5,F9)
         => eventually(A,P5,F3) )
     => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9)) ) ).

% filter_leI
tff(fact_6493_le__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F9: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9))
    <=> ! [P6: fun(A,bool)] :
          ( eventually(A,P6,F9)
         => eventually(A,P6,F3) ) ) ).

% le_filter_def
tff(fact_6494_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,bool)] :
          ( eventually(A,P3,at_infinity(A))
        <=> ? [B7: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B7))
              & ! [X5: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X5)))
                 => pp(aa(A,bool,P3,X5)) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_6495_at__bot__le__at__infinity,axiom,
    pp(aa(filter(real),bool,aa(filter(real),fun(filter(real),bool),ord_less_eq(filter(real)),at_bot(real)),at_infinity(real))) ).

% at_bot_le_at_infinity
tff(fact_6496_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N4),N6))
             => pp(aa(A,bool,P,N4)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6497_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_ym(A,fun(A,bool)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6498_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_yn(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_6499_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G6: filter(A),F9: filter(B),G7: filter(A),F8: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G6)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F3),F9))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G7),G6))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),F8),G7)
           => filterlim(A,B,F8,F9,G7) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6500_le__principal,axiom,
    ! [A: $tType,F3: filter(A),A3: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),principal(A,A3)))
    <=> eventually(A,aTP_Lamp_a(set(A),fun(A,bool),A3),F3) ) ).

% le_principal
tff(fact_6501_filterlim__at__infinity,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: real,F2: fun(C,A),F3: filter(C)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
         => ( filterlim(C,A,F2,at_infinity(A),F3)
          <=> ! [R5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),R5))
               => eventually(C,aa(real,fun(C,bool),aTP_Lamp_yp(fun(C,A),fun(real,fun(C,bool)),F2),R5),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_6502_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A)] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6503_tendsto__sandwich,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),G: fun(B,A),Net: filter(B),H: fun(B,A),C2: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_yq(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),Net)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_yq(fun(B,A),fun(fun(B,A),fun(B,bool)),G),H),Net)
           => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),Net)
             => ( filterlim(B,A,H,topolo7230453075368039082e_nhds(A,C2),Net)
               => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6504_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,at_top(A),F3)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_yr(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F3)
           => filterlim(B,A,G,at_top(A),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6505_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6506_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_ys(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_top
tff(fact_6507_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6508_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_bot
tff(fact_6509_real__tendsto__sandwich,axiom,
    ! [B: $tType,F2: fun(B,real),G: fun(B,real),Net: filter(B),H: fun(B,real),C2: real] :
      ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_yu(fun(B,real),fun(fun(B,real),fun(B,bool)),F2),G),Net)
     => ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_yu(fun(B,real),fun(fun(B,real),fun(B,bool)),G),H),Net)
       => ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(B,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(B,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_6510_eventually__Inf__base,axiom,
    ! [A: $tType,B4: set(filter(A)),P: fun(A,bool)] :
      ( ( B4 != bot_bot(set(filter(A))) )
     => ( ! [F4: filter(A)] :
            ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),F4),B4))
           => ! [G2: filter(A)] :
                ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),G2),B4))
               => ? [X4: filter(A)] :
                    ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X4),B4))
                    & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X4),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),G2))) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4))
        <=> ? [X5: filter(A)] :
              ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X5),B4))
              & eventually(A,P,X5) ) ) ) ) ).

% eventually_Inf_base
tff(fact_6511_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
                 => ( ( ( X5 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),D5)) )
                   => pp(aa(A,bool,P,X5)) ) ) ) ) ) ).

% eventually_at
tff(fact_6512_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X5: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),D5))
                 => pp(aa(A,bool,P,X5)) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_6513_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_yv(fun(A,bool),fun(A,fun(A,bool)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6514_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_yw(fun(B,A),fun(A,fun(B,bool)),F2),L),F3)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_yx(fun(B,A),fun(A,fun(B,bool)),F2),X3),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_6515_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_yy(A,fun(fun(B,A),fun(B,bool)),L),F2),F3)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),X3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_yz(fun(B,A),fun(A,fun(B,bool)),F2),X3),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_6516_filterlim__real__at__infinity__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_infinity(real),at_top(nat)) ).

% filterlim_real_at_infinity_sequentially
tff(fact_6517_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_za(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6518_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_6519_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_zb(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6520_tendsto__le,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: filter(B),F2: fun(B,A),X: A,G: fun(B,A),Y4: A] :
          ( ( F3 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
           => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,Y4),F3)
             => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_zc(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F3)
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) ) ) ) ) ).

% tendsto_le
tff(fact_6521_tendsto__lowerbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_zd(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6522_tendsto__upperbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_ze(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6523_metric__tendsto__imp__tendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C),G: fun(C,B),B2: B] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( eventually(C,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_zf(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),F2),A2),G),B2),F3)
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,B2),F3) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_6524_eventually__at__right__to__0,axiom,
    ! [P: fun(real,bool),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> eventually(real,aa(real,fun(real,bool),aTP_Lamp_zg(fun(real,bool),fun(real,fun(real,bool)),P),A2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_6525_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(B,filter(A)),B4: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F3),B4)))
    <=> ? [X9: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X9),B4))
          & pp(aa(set(B),bool,finite_finite(B),X9))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F3),X9))) ) ) ).

% eventually_INF
tff(fact_6526_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( eventually(A,aTP_Lamp_zh(fun(A,real),fun(A,bool),F2),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zi(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_6527_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
                 => ( ( ( X5 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X5,A2)),D5)) )
                   => pp(aa(A,bool,P,X5)) ) ) ) ) ) ).

% eventually_at_le
tff(fact_6528_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B4: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,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_zj(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C2),N),B4),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6529_tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_zk(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E3),F3) ) ) ) ).

% tendsto_iff
tff(fact_6530_tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_zk(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E),F3) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% tendstoI
tff(fact_6531_tendstoD,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),E2: real] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_zk(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E2),F3) ) ) ) ).

% tendstoD
tff(fact_6532_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,bool),B4: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4))
    <=> ? [X9: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X9),B4))
          & pp(aa(set(filter(A)),bool,finite_finite(filter(A)),X9))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9)) ) ) ).

% eventually_Inf
tff(fact_6533_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_6534_eventually__at__top__to__right,axiom,
    ! [P: fun(real,bool)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_zl(fun(real,bool),fun(real,bool),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_6535_eventually__at__right__to__top,axiom,
    ! [P: fun(real,bool)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_zl(fun(real,bool),fun(real,bool),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_6536_tendsto__arcosh__strong,axiom,
    ! [B: $tType,F2: fun(B,real),A2: real,F3: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),A2))
       => ( eventually(B,aTP_Lamp_zm(fun(B,real),fun(B,bool),F2),F3)
         => filterlim(B,real,aTP_Lamp_zn(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6537_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B3: A] :
                      ( pp(aa(A,bool,Q,B3))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B3),A2)) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6538_tendsto__mult__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( ( C2 != zero_zero(A) )
           => ( filterlim(B,A,G,at_infinity(A),F3)
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zo(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6539_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(C,A),C2: A,F3: filter(C),G: fun(C,A)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(C,A,G,at_infinity(A),F3)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_zp(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_6540_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),N: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_zq(fun(A,B),fun(nat,fun(A,B)),F2),N),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6541_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F3: fun(A,filter(B)),P: fun(B,bool)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),B4))
           => ! [B3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B4))
               => ? [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B4))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F3,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,A4)),aa(A,filter(B),F3,B3)))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F3),B4)))
        <=> ? [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4))
              & eventually(B,P,aa(A,filter(B),F3,X5)) ) ) ) ) ).

% eventually_INF_base
tff(fact_6542_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
               => ( ! [B3: A] :
                      ( pp(aa(A,bool,Q,B3))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B3)) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_6543_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K5: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_zr(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F2),G),K5),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_6544_filterlim__at__withinI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B),A3: set(A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( eventually(B,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_zs(fun(B,A),fun(A,fun(set(A),fun(B,bool))),F2),C2),A3),F3)
           => filterlim(B,A,F2,topolo174197925503356063within(A,C2,A3),F3) ) ) ) ).

% filterlim_at_withinI
tff(fact_6545_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( ( A2 != zero_zero(real) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
              & eventually(A,aTP_Lamp_zt(fun(A,real),fun(A,bool),F2),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ti(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_6546_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_zt(fun(A,real),fun(A,bool),F2),F3)
         => ( 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_ti(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_6547_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_zt(fun(A,real),fun(A,bool),F2),F3)
         => ( 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_ti(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_6548_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( eventually(A,aTP_Lamp_yl(fun(A,real),fun(A,bool),G),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xy(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_6549_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_yl(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_6550_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_yl(fun(A,real),fun(A,bool),F2),F3)
     => ( filterlim(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),F2),at_top(real),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_6551_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_yl(fun(A,real),fun(A,bool),F2),F3)
     => ( filterlim(A,real,F2,at_top(real),F3)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_6552_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_yn(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_6553_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_6554_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_6555_lim__infinity__imp__sequentially,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(real,A),L: A] :
          ( filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(real))
         => filterlim(nat,A,aTP_Lamp_zv(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_6556_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_zw(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F3)
        <=> filterlim(A,B,G,at_infinity(B),F3) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6557_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F8: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),F3,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_6558_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_6559_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_6560_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_6561_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pg(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6562_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F8: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(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_6563_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F8: fun(real,real),Y4: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_6564_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G4: fun(real,real),F8: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F0),F8),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G0),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_6565_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F8: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_6566_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F8: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_6567_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_6568_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F8: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F8),G4),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_6569_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_6570_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_zx(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_6571_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F8: fun(real,real),Y4: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( eventually(real,aTP_Lamp_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_6572_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F8: 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_yi(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F8),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F8),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_yk(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_6573_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y5: B,K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & eventually(A,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_zy(fun(A,B),fun(B,fun(real,fun(A,bool))),F2),Y5),K6),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_6574_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B8: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B8))
               => ~ eventually(A,aa(real,fun(A,bool),aTP_Lamp_zz(fun(A,B),fun(real,fun(A,bool)),F2),B8),F3) ) ) ) ).

% BfunE
tff(fact_6575_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_aaa(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6576_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_aab(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),F2),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_6577_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F3),at_top(nat)))
    <=> ! [N6: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),F3) ) ).

% le_sequentially
tff(fact_6578_eventually__sequentially,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N6: nat] :
        ! [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N4))
         => pp(aa(nat,bool,P,N4)) ) ) ).

% eventually_sequentially
tff(fact_6579_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),X3))
         => pp(aa(nat,bool,P,X3)) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6580_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_6581_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_6582_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_6583_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),K5: real] :
          ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N2))),K5))
         => bfun(nat,A,X7,at_top(nat)) ) ) ).

% BseqI'
tff(fact_6584_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A7: fun(nat,set(A))] :
              ( ! [I4: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I4))
             => ( ! [I4: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I4)))
               => ~ ! [S8: set(A)] :
                      ( topolo1002775350975398744n_open(A,S8)
                     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S8))
                       => eventually(nat,aa(set(A),fun(nat,bool),aTP_Lamp_aad(fun(nat,set(A)),fun(set(A),fun(nat,bool)),A7),S8),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_6585_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_az(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_6586_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_aae(fun(int,A),fun(nat,A),F2),F3,at_top(nat))
     => filterlim(int,A,F2,F3,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_6587_Bseq__eq__bounded,axiom,
    ! [F2: fun(nat,real),A2: real,B2: real] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),aa(set(nat),set(real),image(nat,real,F2),top_top(set(nat)))),set_or1337092689740270186AtMost(real,A2,B2)))
     => bfun(nat,real,F2,at_top(nat)) ) ).

% Bseq_eq_bounded
tff(fact_6588_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aaf(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_6589_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
         => ? [K10: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K10))
              & ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N7))),K10)) ) ) ) ).

% BseqD
tff(fact_6590_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
         => ~ ! [K10: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K10))
               => ~ ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N7))),K10)) ) ) ) ).

% BseqE
tff(fact_6591_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X7: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
         => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N2))),K5))
           => bfun(nat,A,X7,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_6592_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N4))),K6)) ) ) ) ).

% Bseq_def
tff(fact_6593_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [N6: nat] :
            ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% Bseq_iff1a
tff(fact_6594_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [N6: nat] :
            ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X7,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% Bseq_iff
tff(fact_6595_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_6596_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K5: real,F3: filter(A)] :
          ( eventually(A,aa(real,fun(A,bool),aTP_Lamp_zz(fun(A,B),fun(real,fun(A,bool)),F2),K5),F3)
         => bfun(A,B,F2,F3) ) ) ).

% BfunI
tff(fact_6597_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [X5: A] :
                ! [N4: 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,X7,N4)),aa(A,A,uminus_uminus(A),X5)))),K3)) ) ) ) ).

% Bseq_iff2
tff(fact_6598_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( bfun(nat,A,X7,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [N6: nat] :
                ! [N4: 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,X7,N4)),aa(A,A,uminus_uminus(A),aa(nat,A,X7,N6))))),K3)) ) ) ) ).

% Bseq_iff3
tff(fact_6599_Bfun__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( A2 != zero_zero(A) )
           => bfun(B,A,aTP_Lamp_tk(fun(B,A),fun(B,A),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_6600_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & eventually(A,aa(real,fun(A,bool),aTP_Lamp_zz(fun(A,B),fun(real,fun(A,bool)),F2),K6),F3) ) ) ) ).

% Bfun_def
tff(fact_6601_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_aag(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_6602_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aah(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_6603_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_aai(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6604_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_aaj(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_6605_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y6: set(B),X7: set(A),F3: filter(B),F2: fun(A,C)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),Y6)),X7))
     => ( eventually(B,aTP_Lamp_aak(set(B),fun(B,bool),Y6),F3)
       => ( map_filter_on(A,C,X7,F2,map_filter_on(B,A,Y6,G,F3)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_6606_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y: nat] :
            ( pp(aa(nat,bool,P,Y))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_6607_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y: nat] :
            ( pp(aa(nat,bool,P,Y))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_6608_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
     => ( ! [Y: nat] :
            ( pp(aa(nat,bool,P,Y))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_6609_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y: A] :
                ( pp(aa(A,bool,P,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_6610_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y: A] :
                ( pp(aa(A,bool,P,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).

% GreatestI2_order
tff(fact_6611_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [P6: fun(A,bool)] :
                  ( eventually(A,P6,F3)
                  & ! [X5: A,Y5: A] :
                      ( ( pp(aa(A,bool,P6,X5))
                        & pp(aa(A,bool,P6,Y5)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,Y5)),E3)) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_6612_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ( ! [X3: real] :
              ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G) )
           => ( ! [X3: real] :
                  ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_6613_differentiable__cmult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Q2: fun(B,A),C2: A,T2: B] :
          ( differentiable(B,A,aa(A,fun(B,A),aTP_Lamp_aal(fun(B,A),fun(A,fun(B,A)),Q2),C2),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6614_differentiable__cmult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [C2: A,Q2: fun(B,A),T2: B] :
          ( differentiable(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aam(A,fun(fun(B,A),fun(B,A)),C2),Q2),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6615_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ra(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_diff
tff(fact_6616_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X: A,S2: set(A),T2: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
           => differentiable(A,B,F2,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% differentiable_within_subset
tff(fact_6617_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),X: A,S2: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S2))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S2))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aan(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_mult
tff(fact_6618_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S2: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S2))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aao(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% differentiable_divide
tff(fact_6619_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S2: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_aap(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_inverse
tff(fact_6620_nhds__imp__cauchy__filter,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [F3: filter(A),X: A] :
          ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),topolo7230453075368039082e_nhds(A,X)))
         => topolo6773858410816713723filter(A,F3) ) ) ).

% nhds_imp_cauchy_filter
tff(fact_6621_complete__uniform,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A)] :
          ( topolo2479028161051973599mplete(A,S)
        <=> ! [F10: filter(A)] :
              ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F10),principal(A,S)))
             => ( ( F10 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F10)
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
                      & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F10),topolo7230453075368039082e_nhds(A,X5))) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_6622_MVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L2: real,Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
              & has_field_derivative(real,F2,L2,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),L2) ) ) ) ) ) ).

% MVT
tff(fact_6623_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(A),F2: fun(A,B),T2: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
           => topolo81223032696312382ous_on(A,B,T2,F2) ) ) ) ).

% continuous_on_subset
tff(fact_6624_IVT2_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y4: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,B2)),Y4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y4),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F2,X3) = Y4 ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_6625_IVT_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y4: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),Y4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y4),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F2,X3) = Y4 ) ) ) ) ) ) ) ).

% IVT'
tff(fact_6626_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [T2: set(A),G: fun(A,B),S2: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,T2,G)
         => ( topolo81223032696312382ous_on(C,A,S2,F2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S2)),T2))
             => topolo81223032696312382ous_on(C,B,S2,aa(fun(C,A),fun(C,B),aTP_Lamp_aaq(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_6627_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( topolo81223032696312382ous_on(A,B,S2,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => ( aa(A,B,G,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_aar(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6628_continuous__on__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [S2: set(D),F2: fun(D,A),G: fun(D,A)] :
          ( topolo81223032696312382ous_on(D,A,S2,F2)
         => ( topolo81223032696312382ous_on(D,A,S2,G)
           => topolo81223032696312382ous_on(D,A,S2,aa(fun(D,A),fun(D,A),aTP_Lamp_aas(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_6629_continuous__on__mult_H,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [A3: set(D),F2: fun(D,B),G: fun(D,B)] :
          ( topolo81223032696312382ous_on(D,B,A3,F2)
         => ( topolo81223032696312382ous_on(D,B,A3,G)
           => topolo81223032696312382ous_on(D,B,A3,aa(fun(D,B),fun(D,B),aTP_Lamp_aat(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_6630_continuous__on__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(B),F2: fun(B,A),C2: A] :
          ( topolo81223032696312382ous_on(B,A,S2,F2)
         => topolo81223032696312382ous_on(B,A,S2,aa(A,fun(B,A),aTP_Lamp_aau(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_6631_continuous__on__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(B),F2: fun(B,A),C2: A] :
          ( topolo81223032696312382ous_on(B,A,S2,F2)
         => topolo81223032696312382ous_on(B,A,S2,aa(A,fun(B,A),aTP_Lamp_aav(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_6632_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S2: set(A),C2: A] : topolo81223032696312382ous_on(A,A,S2,aa(A,fun(A,A),times_times(A),C2)) ) ).

% continuous_on_mult_const
tff(fact_6633_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real),N: nat] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => topolo81223032696312382ous_on(A,real,S2,aa(nat,fun(A,real),aTP_Lamp_aaw(fun(A,real),fun(nat,fun(A,real)),F2),N)) ) ) ).

% continuous_on_real_root
tff(fact_6634_continuous__on__real__sqrt,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_aax(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_6635_continuous__on__op__minus,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [S2: set(A),X: A] : topolo81223032696312382ous_on(A,A,S2,aa(A,fun(A,A),minus_minus(A),X)) ) ).

% continuous_on_op_minus
tff(fact_6636_continuous__on__diff,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo1633459387980952147up_add(B) )
     => ! [S2: set(D),F2: fun(D,B),G: fun(D,B)] :
          ( topolo81223032696312382ous_on(D,B,S2,F2)
         => ( topolo81223032696312382ous_on(D,B,S2,G)
           => topolo81223032696312382ous_on(D,B,S2,aa(fun(D,B),fun(D,B),aTP_Lamp_aay(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_6637_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S2: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aTP_Lamp_aaz(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_6638_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S2: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aTP_Lamp_aba(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_6639_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,real,F2,X3) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_abb(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_6640_continuous__on__powr,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S2: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S2,F2)
         => ( topolo81223032696312382ous_on(C,real,S2,G)
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),S2))
                 => ( aa(C,real,F2,X3) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(C,real,S2,aa(fun(C,real),fun(C,real),aTP_Lamp_abc(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_6641_continuous__onI__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & dense_order(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( topolo1002775350975398744n_open(B,aa(set(A),set(B),image(A,B,F2),A3))
         => ( ! [X3: A,Y: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) ) ) )
           => topolo81223032696312382ous_on(A,B,A3,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_6642_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( cos(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S2,aTP_Lamp_tl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_6643_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( sin(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S2,aTP_Lamp_tn(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_6644_continuous__on__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A3,F2)
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),A3))
               => ( cosh(A,aa(C,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(C,A,A3,aTP_Lamp_abd(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_6645_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),A3))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,F2,X3))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_abe(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_6646_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ? [C4: real,D3: real] :
            ( ( aa(set(real),set(real),image(real,real,F2),set_or1337092689740270186AtMost(real,A2,B2)) = set_or1337092689740270186AtMost(real,C4,D3) )
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C4),D3)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_6647_continuous__on__arcosh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),A3),set_ord_atLeast(real,one_one(real))))
     => topolo81223032696312382ous_on(real,real,A3,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_6648_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => ? [A7: set(A)] :
              ( topolo1002775350975398744n_open(A,A7)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S2) = collect(A,aa(fun(A,real),fun(A,bool),aTP_Lamp_abf(set(A),fun(fun(A,real),fun(A,bool)),S2),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_6649_continuous__on__powr_H,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S2: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S2,F2)
         => ( topolo81223032696312382ous_on(C,real,S2,G)
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(C,real,F2,X3)))
                    & ( ( aa(C,real,F2,X3) = zero_zero(real) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(C,real,G,X3))) ) ) )
             => topolo81223032696312382ous_on(C,real,S2,aa(fun(C,real),fun(C,real),aTP_Lamp_abc(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_6650_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => ( topolo81223032696312382ous_on(A,real,S2,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,X3))) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                   => ( aa(A,real,F2,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X3))) )
                 => topolo81223032696312382ous_on(A,real,S2,aa(fun(A,real),fun(A,real),aTP_Lamp_abg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6651_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( 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,X3)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_abh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_6652_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( 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,X3)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F2,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_abi(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_6653_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F2: fun(A,A)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
               => ? [Y3: A] : has_field_derivative(A,F2,Y3,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_6654_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F8: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F8,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                & ! [X4: real] : aa(real,real,aa(real,fun(real,real),F8,Z2),X4) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_6655_mvt,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F8: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_derivative(real,real,F2,aa(real,fun(real,real),F8,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),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,A2)) != aa(real,real,aa(real,fun(real,real),F8,Xi),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_6656_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_abj(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_6657_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_abk(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_6658_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_6659_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_6660_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_6661_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),A3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
     => topolo81223032696312382ous_on(real,real,A3,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_6662_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_6663_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_6664_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abl(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).

% ord.Least_def
tff(fact_6665_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,bool),F2: fun(A,B),A3: B,B4: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F2,topolo174197925503356063within(B,A3,B4)))
        <=> ? [S9: set(B)] :
              ( topolo1002775350975398744n_open(B,S9)
              & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S9))
              & ! [X5: A] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X5)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S9),B4)),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))))
                 => pp(aa(A,bool,P,X5)) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_6666_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_6667_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F9: filter(A),F2: fun(B,A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9))
     => pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtercomap(B,A,F2,F3)),filtercomap(B,A,F2,F9))) ) ).

% filtercomap_mono
tff(fact_6668_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F3: filter(B),G6: filter(A)] :
      ( filterlim(A,B,F2,F3,G6)
    <=> pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G6),filtercomap(A,B,F2,F3))) ) ).

% filterlim_iff_le_filtercomap
tff(fact_6669_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,C),F3: fun(B,filter(C)),B4: set(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_abm(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F2),F3)),B4))),filtercomap(A,C,F2,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image(B,filter(C),F3),B4))))) ).

% filtercomap_SUP
tff(fact_6670_eventually__filtercomap__at__top__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_top(A)))
        <=> ? [N6: A] :
            ! [X5: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N6),aa(B,A,F2,X5)))
             => pp(aa(B,bool,P,X5)) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_6671_eventually__filtercomap__at__bot__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_bot(A)))
        <=> ? [N6: A] :
            ! [X5: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X5)),N6))
             => pp(aa(B,bool,P,X5)) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_6672_surj__int__encode,axiom,
    aa(set(int),set(nat),image(int,nat,nat_int_encode),top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_6673_uniformity__dist,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ( topolo7806501430040627800ormity(A) = aa(set(filter(product_prod(A,A))),filter(product_prod(A,A)),complete_Inf_Inf(filter(product_prod(A,A))),aa(set(real),set(filter(product_prod(A,A))),image(real,filter(product_prod(A,A)),aTP_Lamp_abo(real,filter(product_prod(A,A)))),set_ord_greaterThan(real,zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_6674_int__encode__eq,axiom,
    ! [X: int,Y4: int] :
      ( ( aa(int,nat,nat_int_encode,X) = aa(int,nat,nat_int_encode,Y4) )
    <=> ( X = Y4 ) ) ).

% int_encode_eq
tff(fact_6675_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [P6: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),P6,topolo7806501430040627800ormity(A))
             => ? [N6: nat] :
                ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N4))
                 => ! [M5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),M5))
                     => pp(aa(product_prod(A,A),bool,P6,aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,X7,N4)),aa(nat,A,X7,M5)))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_6676_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_6677_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
              & ! [X5: A,Y5: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,Y5)),E3))
                 => pp(aa(product_prod(A,A),bool,P,aa(A,product_prod(A,A),product_Pair(A,A,X5),Y5))) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_6678_uniformity__real__def,axiom,
    topolo7806501430040627800ormity(real) = aa(set(filter(product_prod(real,real))),filter(product_prod(real,real)),complete_Inf_Inf(filter(product_prod(real,real))),aa(set(real),set(filter(product_prod(real,real))),image(real,filter(product_prod(real,real)),aTP_Lamp_abq(real,filter(product_prod(real,real)))),set_ord_greaterThan(real,zero_zero(real)))) ).

% uniformity_real_def
tff(fact_6679_uniformity__complex__def,axiom,
    topolo7806501430040627800ormity(complex) = aa(set(filter(product_prod(complex,complex))),filter(product_prod(complex,complex)),complete_Inf_Inf(filter(product_prod(complex,complex))),aa(set(real),set(filter(product_prod(complex,complex))),image(real,filter(product_prod(complex,complex)),aTP_Lamp_abs(real,filter(product_prod(complex,complex)))),set_ord_greaterThan(real,zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_6680_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),C3: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [T7: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),T7),C3))
               => topolo1002775350975398744n_open(A,aa(B,set(A),F2,T7)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),C3))))
             => ~ ! [C9: set(B)] :
                    ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C9),C3))
                   => ( pp(aa(set(B),bool,finite_finite(B),C9))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),C9)))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_6681_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T11: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T11)))
           => ( ! [B8: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B8),T11))
                 => topolo1002775350975398744n_open(A,B8) )
             => ~ ! [T12: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),T12),T11))
                   => ( pp(aa(set(set(A)),bool,finite_finite(set(A)),T12))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T12))) ) ) ) ) ) ) ).

% compactE
tff(fact_6682_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_6683_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3)) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_6684_compact__diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T6: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( topolo1002775350975398744n_open(A,T6)
           => topolo2193935891317330818ompact(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T6)) ) ) ) ).

% compact_diff
tff(fact_6685_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S2: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S2,F2)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S2))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Xa2))) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_6686_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S2: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S2,F2)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S2))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Xa2)),aa(A,B,F2,X3))) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_6687_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
        <=> ! [C7: set(set(A))] :
              ( ( ! [X5: set(A)] :
                    ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),C7))
                   => topolo1002775350975398744n_open(A,X5) )
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7))) )
             => ? [D8: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),D8),C7))
                  & pp(aa(set(set(A)),bool,finite_finite(set(A)),D8))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D8))) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_6688_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [C8: set(set(A))] :
              ( ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C8))
                 => topolo1002775350975398744n_open(A,X4) )
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8)))
               => ? [C10: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C10),C8))
                    & pp(aa(set(set(A)),bool,finite_finite(set(A)),C10))
                    & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C10))) ) ) )
         => topolo2193935891317330818ompact(A,S2) ) ) ).

% compactI
tff(fact_6689_range__abs__Nats,axiom,
    aa(set(int),set(int),image(int,int,abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_6690_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),I5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
               => topolo7761053866217962861closed(A,aa(B,set(A),F2,I3)) )
           => ( ! [I7: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite(B),I7))
                 => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),I7),I5))
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),I7))) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),I5))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_6691_open__Diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T6: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
         => ( topolo7761053866217962861closed(A,T6)
           => topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T6)) ) ) ) ).

% open_Diff
tff(fact_6692_closed__Diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T6: set(A)] :
          ( topolo7761053866217962861closed(A,S)
         => ( topolo1002775350975398744n_open(A,T6)
           => topolo7761053866217962861closed(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T6)) ) ) ) ).

% closed_Diff
tff(fact_6693_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_6694_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_6695_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_6696_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_6697_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_6698_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_6699_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_6700_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_6701_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_6702_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_6703_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_abt(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% closed_Collect_le
tff(fact_6704_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S: set(A),Y4: A] :
          ( topolo7761053866217962861closed(A,S)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S))
           => ? [U4: set(A),V5: set(A)] :
                ( topolo1002775350975398744n_open(A,U4)
                & topolo1002775350975398744n_open(A,V5)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),U4))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),V5))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ) ).

% t3_space
tff(fact_6705_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S: set(A),T6: set(A)] :
          ( topolo7761053866217962861closed(A,S)
         => ( topolo7761053866217962861closed(A,T6)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T6) = bot_bot(set(A)) )
             => ? [U4: set(A),V5: set(A)] :
                  ( topolo1002775350975398744n_open(A,U4)
                  & topolo1002775350975398744n_open(A,V5)
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),U4))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),V5))
                  & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ) ) ).

% t4_space
tff(fact_6706_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( topolo1002775350975398744n_open(A,A3)
           => ? [A11: set(A)] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A11))
                & topolo7761053866217962861closed(A,A11)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),A3))
                & eventually(A,aTP_Lamp_abu(set(A),fun(A,bool),A11),topolo7230453075368039082e_nhds(A,X)) ) ) ) ) ).

% nhds_closed
tff(fact_6707_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_6708_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_abv(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_6709_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U3: set(A)] :
          ( topolo2193935891317330818ompact(A,U3)
        <=> ! [A14: set(set(A))] :
              ( ! [X5: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),A14))
                 => topolo7761053866217962861closed(A,X5) )
             => ( ! [B10: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B10),A14))
                   => ( pp(aa(set(set(A)),bool,finite_finite(set(A)),B10))
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B10)) != bot_bot(set(A)) ) ) )
               => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A14)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_fip
tff(fact_6710_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [T7: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),T7),F3))
               => topolo7761053866217962861closed(A,T7) )
           => ( ! [F11: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F11))
                 => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F11),F3))
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F11)) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F3)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip
tff(fact_6711_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_pe(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6712_surj__int__decode,axiom,
    aa(set(nat),set(int),image(nat,int,nat_int_decode),top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_6713_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_6714_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_6715_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_6716_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_abw(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6717_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),insert(A,A2),A3))
    <=> ( inj_on(A,B,F2,A3)
        & ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))))) ) ) ).

% inj_on_insert
tff(fact_6718_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ).

% inj_on_diff
tff(fact_6719_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_6720_int__decode__eq,axiom,
    ! [X: nat,Y4: nat] :
      ( ( aa(nat,int,nat_int_decode,X) = aa(nat,int,nat_int_decode,Y4) )
    <=> ( X = Y4 ) ) ).

% int_decode_eq
tff(fact_6721_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_6722_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_on(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6723_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
       => inj_on(A,B,F2,B4) ) ) ).

% inj_on_subset
tff(fact_6724_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
       => inj_on(A,B,F2,A3) ) ) ).

% subset_inj_on
tff(fact_6725_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
                 => ( aa(A,B,F2,X3) != aa(A,B,F2,Y) ) ) ) )
         => ( ! [X3: A,Y: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3)) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_6726_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
         => ( ( aa(set(A),set(B),image(A,B,F2),A3) = aa(set(A),set(B),image(A,B,F2),B4) )
          <=> ( A3 = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_6727_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),aa(set(A),set(B),image(A,B,F2),A3)))
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_6728_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A),T6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(B),set(A),image(B,A,F2),T6)))
    <=> ? [U5: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U5),T6))
          & inj_on(B,A,F2,U5)
          & ( S = aa(set(B),set(A),image(B,A,F2),U5) ) ) ) ).

% subset_image_inj
tff(fact_6729_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4)) ) ) ).

% inj_image_subset_iff
tff(fact_6730_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A8: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A3)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A3)),A8)) )
      <=> ? [G5: fun(B,A)] : aa(set(B),set(A),image(B,A,G5),A8) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_6731_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),image(A,A,F2),A3)))
       => inj_on(A,A,F2,A3) ) ) ).

% finite_surj_inj
tff(fact_6732_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( pp(aa(set(B),bool,finite_finite(B),B4))
         => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ) ).

% inj_on_finite
tff(fact_6733_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,F2),A3)),A3))
       => ( inj_on(A,A,F2,A3)
         => ( aa(set(A),set(A),image(A,A,F2),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_6734_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_6735_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)) ) ) ).

% image_set_diff
tff(fact_6736_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( 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)),A3),B4)),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_6737_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),X: B,B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(A),set(B),image(A,B,F2),A3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),the_inv_into(A,B,A3,F2,X)),B4)) ) ) ) ).

% the_inv_into_into
tff(fact_6738_injective__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C2: real] :
          ( ( C2 != zero_zero(real) )
         => inj_on(A,A,real_V8093663219630862766scaleR(A,C2),top_top(set(A))) ) ) ).

% injective_scaleR
tff(fact_6739_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C)] :
      ( ! [I3: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I3)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
           => inj_on(B,C,F2,aa(A,set(B),A3,I3)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_6740_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S: set(A),T6: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),S))
     => ( pp(aa(set(B),bool,finite_finite(B),T6))
       => ( ( aa(set(A),nat,finite_card(A),S) = aa(set(B),nat,finite_card(B),T6) )
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),S)),T6))
           => ( ! [X5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),T6))
                 => ? [Xa4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S))
                      & ( aa(A,B,F2,Xa4) = X5 ) ) )
            <=> inj_on(A,B,F2,S) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_6741_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( inj_on(B,A,G,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B4)),A3))
           => ( pp(aa(set(A),bool,finite_finite(A),A3))
             => ( pp(aa(set(B),bool,finite_finite(B),B4))
               => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B4) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_6742_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,F2),A3)))) ) ).

% inj_image_Compl_subset
tff(fact_6743_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),C3: set(A),A3: set(C),B4: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F2,C3)
     => ( ! [X3: C] :
            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),A3))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B4,X3)),C3)) )
       => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J),A3))
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),A3))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_abx(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B4)),A3)) ) ) ) ) ).

% image_INT
tff(fact_6744_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_6745_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B4))
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A3)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A3)),B4)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_6746_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( pp(aa(set(B),bool,finite_finite(B),B4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4))) ) ) ) ).

% card_inj_on_le
tff(fact_6747_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)))
         => ? [F5: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F5),A3)),B4))
              & inj_on(A,B,F5,A3) ) ) ) ) ).

% card_le_inj
tff(fact_6748_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,log(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_6749_complex__is__Nat__iff,axiom,
    ! [Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Z),semiring_1_Nats(complex)))
    <=> ( ( im(Z) = zero_zero(real) )
        & ? [I2: nat] : re(Z) = aa(nat,real,semiring_1_of_nat(real),I2) ) ) ).

% complex_is_Nat_iff
tff(fact_6750_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( inj_on(B,A,G,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B4)),A3))
           => ? [H3: fun(A,B)] : bij_betw(A,B,H3,A3,B4) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_6751_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
          & inj_on(B,A,F2,T9)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% ex_subset_image_inj
tff(fact_6752_inj__int__encode,axiom,
    ! [A3: set(int)] : inj_on(int,nat,nat_int_encode,A3) ).

% inj_int_encode
tff(fact_6753_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_6754_inj__prod__encode,axiom,
    ! [A3: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A3) ).

% inj_prod_encode
tff(fact_6755_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N5: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N5) ) ).

% inj_on_of_nat
tff(fact_6756_inj__Suc,axiom,
    ! [N5: set(nat)] : inj_on(nat,nat,suc,N5) ).

% inj_Suc
tff(fact_6757_inj__int__decode,axiom,
    ! [A3: set(nat)] : inj_on(nat,int,nat_int_decode,A3) ).

% inj_int_decode
tff(fact_6758_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_6759_inj__on__diff__nat,axiom,
    ! [N5: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_ns(nat,fun(nat,nat),K),N5) ) ).

% inj_on_diff_nat
tff(fact_6760_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X7: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X7),Y6))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X7,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X7,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_6761_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S))
    <=> ? [F7: fun(nat,A)] :
          ( inj_on(nat,A,F7,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F7),top_top(set(nat)))),S)) ) ) ).

% infinite_iff_countable_subset
tff(fact_6762_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S))
     => ? [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)),aa(set(nat),set(A),image(nat,A,F5),top_top(set(nat)))),S)) ) ) ).

% infinite_countable_subset
tff(fact_6763_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3)))
         => summable(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G)) ) ) ) ).

% summable_reindex
tff(fact_6764_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A),S2: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),S2) = S2 )
     => ( ! [M4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M4),F2),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_aby(fun(A,A),fun(A,fun(nat,A)),F2),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_6765_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G))),suminf(real,F2))) ) ) ) ).

% suminf_reindex_mono
tff(fact_6766_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_6767_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3)))
         => ( ! [X3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat)))))
               => ( aa(nat,real,F2,X3) = zero_zero(real) ) )
           => ( suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_6768_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S)))
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
            & inj_on(B,A,F2,T9) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% all_subset_image_inj
tff(fact_6769_has__derivative__power__int,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),X: C,F8: fun(C,A),S: set(C),N: int] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F8,topolo174197925503356063within(C,X,S))
           => has_derivative(C,A,aa(int,fun(C,A),aTP_Lamp_abz(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_aca(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),F2),X),F8),N),topolo174197925503356063within(C,X,S)) ) ) ) ).

% has_derivative_power_int
tff(fact_6770_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,N: int,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_acb(int,fun(A,A),N),aa(int,fun(A,A),aTP_Lamp_acc(A,fun(int,fun(A,A)),X),N),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_power_int'
tff(fact_6771_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_6772_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y4: A] : power_int(A,Y4,one_one(int)) = Y4 ) ).

% power_int_1_right
tff(fact_6773_power__int__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,N: int] : aa(A,A,sgn_sgn(A),power_int(A,A2,N)) = power_int(A,aa(A,A,sgn_sgn(A),A2),N) ) ).

% power_int_sgn
tff(fact_6774_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W)),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,aa(num,A,numeral_numeral(A),W),M)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_6775_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W: num,Y4: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y4),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),M)),power_int(A,Y4,M)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_6776_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_6777_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          ( ( M != zero_zero(int) )
         => ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_6778_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_6779_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,N: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A2),N)) = aa(A,A,abs_abs(A),power_int(A,A2,N)) ) ).

% abs_power_int_minus
tff(fact_6780_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_6781_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M)),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),M),N))) ) ).

% power_int_mult_numeral
tff(fact_6782_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_6783_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: 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)),M)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_6784_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_6785_numeral__power__int__eq__of__real__cancel__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: num,N: int,Y4: real] :
          ( ( power_int(A,aa(num,A,numeral_numeral(A),X),N) = real_Vector_of_real(A,Y4) )
        <=> ( power_int(real,aa(num,real,numeral_numeral(real),X),N) = Y4 ) ) ) ).

% numeral_power_int_eq_of_real_cancel_iff
tff(fact_6786_of__real__eq__numeral__power__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Y4: real,X: num,N: int] :
          ( ( real_Vector_of_real(A,Y4) = power_int(A,aa(num,A,numeral_numeral(A),X),N) )
        <=> ( Y4 = power_int(real,aa(num,real,numeral_numeral(real),X),N) ) ) ) ).

% of_real_eq_numeral_power_int_cancel_iff
tff(fact_6787_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y4: A] : power_int(A,Y4,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y4) ) ).

% power_int_minus1_right
tff(fact_6788_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: 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),M))),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),M),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_6789_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),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),M),N))) ) ).

% power_int_add_numeral
tff(fact_6790_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( 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,A2,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_6791_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A2: 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),A2),N) = power_int(A,A2,N) ) ) ) ).

% power_int_minus_left_even
tff(fact_6792_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A2: 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),A2),N) = aa(A,A,uminus_uminus(A),power_int(A,A2,N)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_6793_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: num,N: num] : power_int(A,aa(num,A,numeral_numeral(A),M),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(M,N))) ) ).

% power_int_numeral_neg_numeral
tff(fact_6794_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_6795_power__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,N: int] : aa(A,A,abs_abs(A),power_int(A,A2,N)) = power_int(A,aa(A,A,abs_abs(A),A2),N) ) ).

% power_int_abs
tff(fact_6796_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_6797_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y4: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y4),M) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,Y4,M)) ) ).

% power_int_divide_distrib
tff(fact_6798_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = power_int(A,power_int(A,X,M),N) ) ).

% power_int_mult
tff(fact_6799_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_6800_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y4: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y4),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y4,M)) ) ).

% power_int_mult_distrib
tff(fact_6801_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_6802_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_6803_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_6804_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_6805_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F2: fun(A,B),N: int] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aa(int,fun(A,B),aTP_Lamp_acd(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_on_power_int
tff(fact_6806_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          ( ( ( M = zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = one_one(A) ) )
          & ( ( M != zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ) ).

% power_int_0_left_If
tff(fact_6807_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A2,N)),power_int(A,A2,N5))) ) ) ) ).

% power_int_increasing
tff(fact_6808_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A2,N)),power_int(A,A2,N5))) ) ) ) ).

% power_int_strict_increasing
tff(fact_6809_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_6810_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != N ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_diff
tff(fact_6811_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A2),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A2)) ) ).

% power_int_minus_one_diff_commute
tff(fact_6812_tendsto__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),N: int] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( A2 != zero_zero(A) )
           => filterlim(B,A,aa(int,fun(B,A),aTP_Lamp_ace(fun(B,A),fun(int,fun(B,A)),F2),N),topolo7230453075368039082e_nhds(A,power_int(A,A2,N)),F3) ) ) ) ).

% tendsto_power_int
tff(fact_6813_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S2: set(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),aa(int,fun(A,B),aTP_Lamp_acf(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_6814_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S2: set(A),N: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_acg(fun(A,B),fun(int,fun(A,B)),F2),N),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_power_int
tff(fact_6815_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_xb(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aa(int,fun(A,B),aTP_Lamp_acf(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_power_int
tff(fact_6816_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A2,N5)),power_int(A,A2,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_6817_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y4: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( 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,Y4,N))) ) ) ) ) ).

% power_int_mono
tff(fact_6818_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( 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,A2,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_6819_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_6820_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( 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,A2,N))) ) ) ) ).

% one_less_power_int
tff(fact_6821_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_6822_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( division_ring(A)
        & one(B)
        & uminus(B) )
     => ! [X: C,A2: 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),A2),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,A2,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_6823_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( 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,A2,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_6824_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( 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,A2,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_6825_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_6826_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N5 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A2,N5)),power_int(A,A2,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_6827_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: 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,M)),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),M),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_6828_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: 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,M)),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),M),N)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_6829_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int,A2: 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),A2),N) = power_int(A,A2,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),A2),N) = aa(A,A,uminus_uminus(A),power_int(A,A2,N)) ) ) ) ) ).

% power_int_minus_left
tff(fact_6830_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_6831_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).

% power_int_add_1
tff(fact_6832_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).

% power_int_add_1'
tff(fact_6833_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_6834_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)) )
       => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = power_int(real,X,N) ) ) ) ).

% powr_real_of_int'
tff(fact_6835_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S2: set(A),N: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_ach(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,S2)) ) ) ) ).

% DERIV_power_int
tff(fact_6836_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 )
                & ! [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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_6837_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 )
                  & ! [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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_6838_ball__empty,axiom,
    ! [A: $tType,P: fun(A,bool),X4: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),bot_bot(set(A))))
     => pp(aa(A,bool,P,X4)) ) ).

% ball_empty
tff(fact_6839_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),top_top(set(A))))
         => pp(aa(A,bool,P,X5)) )
    <=> ! [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).

% ball_UNIV
tff(fact_6840_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_acj(product_prod(A,A),bool))) ) ).

% closed_subdiagonal
tff(fact_6841_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_ack(product_prod(A,A),bool))) ) ).

% closed_superdiagonal
tff(fact_6842_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : collect(A,aTP_Lamp_acl(fun(B,A),fun(A,bool),F2)) = aa(set(B),set(A),image(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_6843_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,bool)] : collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_acm(fun(B,A),fun(fun(B,bool),fun(A,bool)),F2),P)) = aa(set(B),set(A),image(B,A,F2),collect(B,P)) ).

% setcompr_eq_image
tff(fact_6844_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : collect(A,aa(set(B),fun(A,bool),aTP_Lamp_acn(fun(B,A),fun(set(B),fun(A,bool)),F2),A3)) = aa(set(B),set(A),image(B,A,F2),A3) ).

% Setcompr_eq_image
tff(fact_6845_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A3) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_aco(set(A),fun(A,bool),A3))) ) ).

% Sup_eq_Inf
tff(fact_6846_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A3) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_acp(set(A),fun(A,bool),A3))) ) ).

% Inf_eq_Sup
tff(fact_6847_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 )
        & ! [X5: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
           => vEBT_VEBT_valid(X5,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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList),Summary)),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_6848_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),X: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_acq(fun(A,A),fun(A,fun(A,bool)),P3),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),P3),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_6849_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y4) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y4)
          <=> ( 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(Y4)
              <=> ~ ( ( Deg2 = Xa )
                    & ! [X5: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X5,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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_6850_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(aa(product_prod(vEBT_VEBT,nat),bool,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) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_6851_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_valid(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(aa(product_prod(vEBT_VEBT,nat),bool,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) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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 )
                      & ! [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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_6852_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_acr(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% Inf_Sup_le
tff(fact_6853_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_acs(set(set(A)),fun(set(A),bool),A3))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3)))) ) ).

% Sup_Inf_le
tff(fact_6854_Pow__Compl,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = collect(set(A),aTP_Lamp_act(set(A),fun(set(A),bool),A3)) ).

% Pow_Compl
tff(fact_6855_Sup__int__def,axiom,
    ! [X7: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X7) = the(int,aTP_Lamp_acu(set(int),fun(int,bool),X7)) ).

% Sup_int_def
tff(fact_6856_Inf__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),collect(filter(A),aTP_Lamp_acv(set(filter(A)),fun(filter(A),bool),S))) ).

% Inf_filter_def
tff(fact_6857_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_acw(A,bool)) ) ) ).

% Nats_altdef1
tff(fact_6858_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y4: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y4) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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(Y4)
                <=> ( Xa = one_one(nat) ) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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(Y4)
                  <=> ( ( Deg2 = Xa )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X5,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(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_aci(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) )
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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_6859_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_acx(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% finite_Inf_Sup
tff(fact_6860_Ball__Collect,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
         => pp(aa(A,bool,P,X5)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,P))) ) ).

% Ball_Collect
tff(fact_6861_refl__ge__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] :
      ( ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),R,X3),X3))
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R)) ) ).

% refl_ge_eq
tff(fact_6862_ge__eq__refl,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X: A] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R))
     => pp(aa(A,bool,aa(A,fun(A,bool),R,X),X)) ) ).

% ge_eq_refl
tff(fact_6863_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F2,R) = collect(product_prod(A,A),product_case_prod(A,A,bool,aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_acy(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F2),R))) ).

% mlex_eq
tff(fact_6864_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_acz(real,bool)) ).

% Rats_eq_int_div_nat
tff(fact_6865_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_6866_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_6867_Rats__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_diff
tff(fact_6868_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_6869_Rats__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),A2),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_divide
tff(fact_6870_Rats__number__of,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),field_char_0_Rats(A))) ) ).

% Rats_number_of
tff(fact_6871_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),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),times_times(A),A2),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_mult
tff(fact_6872_Rats__dense__in__real,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4))
     => ? [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),Y4)) ) ) ).

% Rats_dense_in_real
tff(fact_6873_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),X)) ) ).

% Rats_no_bot_less
tff(fact_6874_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),X3)) ) ).

% Rats_no_top_le
tff(fact_6875_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A))) ) ).

% Ints_subset_Rats
tff(fact_6876_Nats__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),field_char_0_Rats(A))) ) ).

% Nats_subset_Rats
tff(fact_6877_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y4: A,R: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y4)))
     => ( 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),Y4)),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),Y4)),mlex_prod(A,F2,R))) ) ) ).

% mlex_leq
tff(fact_6878_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_ada(real,bool)) ).

% Rats_eq_int_div_int
tff(fact_6879_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)),aa(set(nat),set(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)),aa(set(nat),set(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_6880_rat__floor__lemma,axiom,
    ! [A2: 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),A2),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),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),A2),B2)),one_one(int))))) ) ).

% rat_floor_lemma
tff(fact_6881_Un__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),sup_sup(set(A)),A3),B4)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% Un_iff
tff(fact_6882_UnCI,axiom,
    ! [A: $tType,C2: A,B4: set(A),A3: set(A)] :
      ( ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))) ) ).

% UnCI
tff(fact_6883_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ) ) ).

% le_sup_iff
tff(fact_6884_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.bounded_iff
tff(fact_6885_Un__empty,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> ( ( A3 = bot_bot(set(A)) )
        & ( B4 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_6886_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: 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)),sup_sup(set(A)),A3),B4)),C3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3)) ) ) ).

% Un_subset_iff
tff(fact_6887_Un__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) ).

% Un_insert_right
tff(fact_6888_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A2),B4)),C3) = aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) ).

% Un_insert_left
tff(fact_6889_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T6: 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)),sup_sup(set(A)),S),T6)),S) = S ).

% Un_Int_eq(1)
tff(fact_6890_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T6: 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)),sup_sup(set(A)),S),T6)),T6) = T6 ).

% Un_Int_eq(2)
tff(fact_6891_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T6)) = S ).

% Un_Int_eq(3)
tff(fact_6892_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T6: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T6)) = T6 ).

% Un_Int_eq(4)
tff(fact_6893_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T6: 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)),S),T6)),S) = S ).

% Int_Un_eq(1)
tff(fact_6894_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T6: 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)),S),T6)),T6) = T6 ).

% Int_Un_eq(2)
tff(fact_6895_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T6)) = S ).

% Int_Un_eq(3)
tff(fact_6896_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T6: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T6)) = T6 ).

% Int_Un_eq(4)
tff(fact_6897_Un__Diff__cancel2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),A3) ).

% Un_Diff_cancel2
tff(fact_6898_Un__Diff__cancel,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) ).

% Un_Diff_cancel
tff(fact_6899_Compl__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),B4) ).

% Compl_Diff_eq
tff(fact_6900_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool),F2: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_adb(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),collect(B,P)))),aa(set(B),set(A),image(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),collect(B,aTP_Lamp_adc(fun(B,bool),fun(B,bool),P))))) ).

% if_image_distrib
tff(fact_6901_mult__rat,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] : aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),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),A2),C2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ).

% mult_rat
tff(fact_6902_divide__rat,axiom,
    ! [A2: 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,A2),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),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) ).

% divide_rat
tff(fact_6903_floor__Fract,axiom,
    ! [A2: int,B2: int] : archim6421214686448440834_floor(rat,aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) ).

% floor_Fract
tff(fact_6904_less__rat,axiom,
    ! [B2: int,D2: int,A2: 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,A2),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),A2),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_6905_add__rat,axiom,
    ! [B2: int,D2: int,A2: 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,A2),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),A2),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_6906_le__rat,axiom,
    ! [B2: int,D2: int,A2: 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,A2),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),A2),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_6907_diff__rat,axiom,
    ! [B2: int,D2: int,A2: 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,A2),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),A2),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_6908_sgn__rat,axiom,
    ! [A2: int,B2: int] : aa(rat,rat,sgn_sgn(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A2)),aa(int,int,sgn_sgn(int),B2))) ).

% sgn_rat
tff(fact_6909_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6910_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_6911_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: 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)),A3),B4)),C3))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3))) ) ).

% Diff_subset_conv
tff(fact_6912_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = B4 ) ) ).

% Diff_partition
tff(fact_6913_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)))) ) ).

% distrib_inf_le
tff(fact_6914_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y4: 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),Y4),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)))) ) ).

% distrib_sup_le
tff(fact_6915_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_6916_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y4: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4))) ) ).

% inf_sup_ord(4)
tff(fact_6917_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y4: 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),Y4))) ) ).

% inf_sup_ord(3)
tff(fact_6918_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: 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),A2),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_6919_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),X)) ) ) ) ).

% le_supI
tff(fact_6920_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y4: 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),Y4))) ) ).

% sup_ge1
tff(fact_6921_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y4: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4))) ) ).

% sup_ge2
tff(fact_6922_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => 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),A2),B2))) ) ) ).

% le_supI1
tff(fact_6923_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: 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),A2),B2))) ) ) ).

% le_supI2
tff(fact_6924_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( 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),A2),B2))) ) ) ) ).

% sup.mono
tff(fact_6925_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2))) ) ) ) ).

% sup_mono
tff(fact_6926_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y4: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),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),Y4),Z)),X)) ) ) ) ).

% sup_least
tff(fact_6927_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4) = Y4 ) ) ) ).

% le_iff_sup
tff(fact_6928_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_6929_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% sup.orderI
tff(fact_6930_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y4: A] :
          ( ! [X3: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F2,X3),Y)))
         => ( ! [X3: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),F2,X3),Y)))
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y),Z2)),X3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4) = aa(A,A,aa(A,fun(A,A),F2,X),Y4) ) ) ) ) ) ).

% sup_unique
tff(fact_6931_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_6932_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_6933_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4) = X ) ) ) ).

% sup_absorb1
tff(fact_6934_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4) = Y4 ) ) ) ).

% sup_absorb2
tff(fact_6935_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: 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)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.boundedE
tff(fact_6936_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => 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)),A2)) ) ) ) ).

% sup.boundedI
tff(fact_6937_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_6938_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ).

% sup.cobounded1
tff(fact_6939_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: 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),A2),B2))) ) ).

% sup.cobounded2
tff(fact_6940_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_6941_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_6942_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => 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),A2),B2))) ) ) ).

% sup.coboundedI1
tff(fact_6943_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: 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),A2),B2))) ) ) ).

% sup.coboundedI2
tff(fact_6944_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_6945_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))
     => ~ ! [A11: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),A3))
           => ! [B13: set(A)] :
                ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B13),B4))
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A11),B13) ) ) ) ) ).

% subset_UnE
tff(fact_6946_Un__absorb2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = A3 ) ) ).

% Un_absorb2
tff(fact_6947_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_6948_Un__upper2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))) ).

% Un_upper2
tff(fact_6949_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))) ).

% Un_upper1
tff(fact_6950_Un__least,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C3))
       => 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)),sup_sup(set(A)),A3),B4)),C3)) ) ) ).

% Un_least
tff(fact_6951_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B4: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),D4))
       => 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)),sup_sup(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D4))) ) ) ).

% Un_mono
tff(fact_6952_Un__Pow__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),pow2(A,B4))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))) ).

% Un_Pow_subset
tff(fact_6953_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_6954_Compl__partition2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = top_top(set(A)) ).

% Compl_partition2
tff(fact_6955_Compl__partition,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = top_top(set(A)) ).

% Compl_partition
tff(fact_6956_Un__UNIV__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B4) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_6957_Un__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_6958_Rat__induct__pos,axiom,
    ! [P: fun(rat,bool),Q2: rat] :
      ( ! [A4: int,B3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
         => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B3))) )
     => pp(aa(rat,bool,P,Q2)) ) ).

% Rat_induct_pos
tff(fact_6959_Fract__coprime,axiom,
    ! [A2: int,B2: int] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2))) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ).

% Fract_coprime
tff(fact_6960_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_6961_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_6962_eq__rat_I2_J,axiom,
    ! [A2: int] : aa(int,rat,aa(int,fun(int,rat),fract,A2),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% eq_rat(2)
tff(fact_6963_mult__rat__cancel,axiom,
    ! [C2: int,A2: 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),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ) ).

% mult_rat_cancel
tff(fact_6964_eq__rat_I1_J,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( ( aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) = aa(int,rat,aa(int,fun(int,rat),fract,C2),D2) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A2),D2) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_6965_Un__left__commute,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_left_commute
tff(fact_6966_Un__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) ).

% Un_left_absorb
tff(fact_6967_Un__commute,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),A3) ).

% Un_commute
tff(fact_6968_Un__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),A3) = A3 ).

% Un_absorb
tff(fact_6969_Un__assoc,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: 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)),sup_sup(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) ).

% Un_assoc
tff(fact_6970_ball__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),P: fun(A,bool)] :
      ( ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))
         => pp(aa(A,bool,P,X5)) )
    <=> ( ! [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
           => pp(aa(A,bool,P,X5)) )
        & ! [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4))
           => pp(aa(A,bool,P,X5)) ) ) ) ).

% ball_Un
tff(fact_6971_bex__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),P: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))
          & pp(aa(A,bool,P,X5)) )
    <=> ( ? [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
            & pp(aa(A,bool,P,X5)) )
        | ? [X5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B4))
            & pp(aa(A,bool,P,X5)) ) ) ) ).

% bex_Un
tff(fact_6972_UnI2,axiom,
    ! [A: $tType,C2: A,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4))
     => 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)),sup_sup(set(A)),A3),B4))) ) ).

% UnI2
tff(fact_6973_UnI1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))) ) ).

% UnI1
tff(fact_6974_UnE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: 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)),sup_sup(set(A)),A3),B4)))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% UnE
tff(fact_6975_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_add(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)),collect(A,P)),collect(A,Q)) ).

% Collect_disj_eq
tff(fact_6976_Un__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ade(set(A),fun(set(A),fun(A,bool)),A3),B4)) ).

% Un_def
tff(fact_6977_eq__rat_I3_J,axiom,
    ! [A2: int,C2: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),A2) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),C2) ).

% eq_rat(3)
tff(fact_6978_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_adf(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_6979_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C3)) ).

% Un_Diff
tff(fact_6980_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_6981_Un__Int__distrib2,axiom,
    ! [A: $tType,B4: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)),A3) = 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)),sup_sup(set(A)),B4),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_distrib2
tff(fact_6982_Int__Un__distrib2,axiom,
    ! [A: $tType,B4: set(A),C3: set(A),A3: 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)),sup_sup(set(A)),B4),C3)),A3) = 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)),B4),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) ).

% Int_Un_distrib2
tff(fact_6983_Un__Int__distrib,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) = 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)),sup_sup(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_Int_distrib
tff(fact_6984_Int__Un__distrib,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_Un_distrib
tff(fact_6985_Un__Int__crazy,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: 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)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) = 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)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_crazy
tff(fact_6986_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Un
tff(fact_6987_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Int
tff(fact_6988_Int__Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = A3 ).

% Int_Diff_Un
tff(fact_6989_Un__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = A3 ).

% Un_Diff_Int
tff(fact_6990_Compl__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = 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)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Compl_Un
tff(fact_6991_Compl__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Compl_Int
tff(fact_6992_insert__def,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_lu(A,fun(A,bool),A2))),B4) ).

% insert_def
tff(fact_6993_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( ( 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)),sup_sup(set(A)),A3),B4) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_6994_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_6995_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_6996_Un__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).

% Un_empty_right
tff(fact_6997_Un__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B4) = B4 ).

% Un_empty_left
tff(fact_6998_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B4: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B4)) ).

% image_Un
tff(fact_6999_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y4) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y4)) ) ) ).

% sup_shunt
tff(fact_7000_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q2: A,R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q2))),R2)) ) ) ).

% sup_neg_inf
tff(fact_7001_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: 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),Y4))),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),Y4),Z))) ) ) ).

% shunt2
tff(fact_7002_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4)),Z))) ) ) ).

% shunt1
tff(fact_7003_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(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(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)))) ) ).

% less_eq_Inf_inter
tff(fact_7004_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7005_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_7006_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B4: set(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)),sup_sup(set(A)),A3),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)))) ).

% card_Un_le
tff(fact_7007_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_7008_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7009_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_7010_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_7011_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_7012_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_7013_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_7014_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_7015_Pow__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : pow2(A,aa(set(A),set(A),insert(A,A2),A3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,A2)),pow2(A,A3))) ).

% Pow_insert
tff(fact_7016_Inter__Un__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B4: set(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)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B4)))) ).

% Inter_Un_subset
tff(fact_7017_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7018_prod_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,G,B4)) ) ) ) ) ).

% prod.union_inter
tff(fact_7019_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_7020_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_7021_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_7022_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_7023_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% zero_less_Fract_iff
tff(fact_7024_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_7025_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_7026_Fract__less__one__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_7027_one__less__Fract__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A2)) ) ) ).

% one_less_Fract_iff
tff(fact_7028_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_om(A,fun(nat,A),B4)),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4) ) ).

% SUP_nat_binary
tff(fact_7029_Fract__add__one,axiom,
    ! [N: int,M: 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),M),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M),N)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_7030_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_adg(A,fun(A,bool)),aTP_Lamp_adh(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_7031_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4)))
                 => ( aa(B,A,G,X3) = 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)),A3),B4)) = 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),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7032_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( 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)),A3),B4)) = 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),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B4))),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)),A3),B4))) ) ) ) ) ).

% sum_Un
tff(fact_7033_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,G,B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7034_prod_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4) = bot_bot(set(B)) )
             => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,G,B4)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7035_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( 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)),A3),B4)) = 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)),A3),B4))),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)),B4),A3)))),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)),A3),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_7036_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: 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)),A3),B4)))
         => ( 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)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,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)),A3),B4))),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)),B4),A3)))),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)),A3),B4))) ) ) ) ).

% sum_Un2
tff(fact_7037_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7038_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4))),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A3)))),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))) ) ) ) ) ).

% prod.union_diff2
tff(fact_7039_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))
    <=> ( inj_on(A,B,F2,A3)
        & inj_on(A,B,F2,B4)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_7040_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_7041_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_7042_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B4))
       => ( 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)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B4))),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)),A3),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_7043_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_7044_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_7045_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% zero_le_Fract_iff
tff(fact_7046_one__le__Fract__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2)) ) ) ).

% one_le_Fract_iff
tff(fact_7047_Fract__le__one__iff,axiom,
    ! [B2: int,A2: 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,A2),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_7048_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_7049_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_7050_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4)))
                 => ( aa(B,A,F2,X3) != zero_zero(A) ) )
             => ( groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,F2,B4))),groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_7051_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ys: list(B)] :
          ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
         => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
           => ( distinct(A,map(B,A,F2,Xs))
             => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Ys))
               => ( distinct(A,map(B,A,F2,Ys))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7052_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B4: set(A),X: A] :
      ( inj_on(A,B,G,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))))
       => pp(aa(set(fun(B,A)),bool,aa(fun(B,A),fun(set(fun(B,A)),bool),member(fun(B,A)),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_adi(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_7053_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I: B,B4: set(A),J4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),fun_upd(B,set(A),A3,I,B4)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))))),if(set(A),aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),J4),B4,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_7054_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F12: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtercomap(A,B,F2,F12)),filtercomap(A,B,F2,F23))),filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F23)))) ).

% filtercomap_sup
tff(fact_7055_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ).

% sup_set_def
tff(fact_7056_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_7057_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_7058_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_7059_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,X: B,A3: set(B),F2: fun(B,A),Y4: A] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => ( aa(set(B),set(A),image(B,A,fun_upd(B,A,F2,X,Y4)),A3) = aa(set(A),set(A),insert(A,Y4),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),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),A3))
       => ( aa(set(B),set(A),image(B,A,fun_upd(B,A,F2,X,Y4)),A3) = aa(set(B),set(A),image(B,A,F2),A3) ) ) ) ).

% fun_upd_image
tff(fact_7060_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A15: set(B),B14: set(A),F22: fun(C,D),B23: set(C),A25: set(D)] :
      ( ( aa(set(B),set(A),image(B,A,F1),A15) = B14 )
     => ( inj_on(C,D,F22,B23)
       => ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image(C,D,F22),B23)),A25))
         => ( ( ( B23 = bot_bot(set(C)) )
             => ( A25 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B23,B14) = aa(set(fun(D,B)),set(fun(C,A)),image(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A25,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_7061_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F8: fun(real,real)] :
      ( ! [X3: real] : has_field_derivative(real,F2,aa(real,real,F8,X3),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( ! [X3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F8,X3)))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_7062_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y4)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ).

% strict_mono_less_eq
tff(fact_7063_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R2: fun(A,B),M: A,N: A] :
          ( order_strict_mono(A,B,R2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,R2,M)),aa(A,B,R2,N))) ) ) ) ).

% strict_mono_leD
tff(fact_7064_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y4))) ) ) ) ).

% strict_monoD
tff(fact_7065_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_7066_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X5: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X5)),aa(A,B,F2,Y5))) ) ) ) ).

% strict_mono_def
tff(fact_7067_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: 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,Y4)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ) ).

% strict_mono_less
tff(fact_7068_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( order_strict_mono(A,B,F2)
         => ( ( aa(A,B,F2,X) = aa(A,B,F2,Y4) )
          <=> ( X = Y4 ) ) ) ) ).

% strict_mono_eq
tff(fact_7069_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A25: set(A),A15: set(B),F1: fun(B,C),B14: set(C),F22: fun(D,A),B23: set(D)] :
      ( pp(aa(set(fun(A,B)),bool,aa(fun(A,B),fun(set(fun(A,B)),bool),member(fun(A,B)),G),bNF_Wellorder_Func(A,B,A25,A15)))
     => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,F1),A15)),B14))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image(D,A,F22),B23)),A25))
         => pp(aa(set(fun(D,C)),bool,aa(fun(D,C),fun(set(fun(D,C)),bool),member(fun(D,C)),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B23,F1,F22),G)),bNF_Wellorder_Func(D,C,B23,B14))) ) ) ) ).

% Func_map
tff(fact_7070_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_ng(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_7071_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_adj(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_7072_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_7073_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_7074_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_7075_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_7076_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_7077_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = some(num,one2) ).

% take_bit_num_simps(2)
tff(fact_7078_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = some(num,one2) ).

% take_bit_num_simps(5)
tff(fact_7079_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_7080_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_7081_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4)))) ) ) ).

% strict_mono_Suc_iff
tff(fact_7082_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num,Q2: num] :
          ( ( bit_take_bit_num(M,N) = some(num,Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_7083_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_7084_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_adk(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_7085_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_ni(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_int_def
tff(fact_7086_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_7087_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_nk(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_7088_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_7089_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),aa(set(nat),set(nat),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_adl(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_7090_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),aa(set(nat),set(nat),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_adl(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C2)
            <=> sums(A,F2,C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_7091_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),aa(set(nat),set(nat),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_adm(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_7092_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = some(num,num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_7093_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X3: nat,Y: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,X3))),real_V7770717601297561774m_norm(A,aa(nat,A,F2,Y)))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_adn(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat))
            <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_7094_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_7095_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_7096_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_7097_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_ado(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_7098_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_7099_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_ado(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_7100_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_7101_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_7102_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_7103_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_7104_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_7105_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit0,M),one2) = some(num,aa(num,num,bit0,M)) ).

% and_not_num.simps(4)
tff(fact_7106_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit0,N)) = some(num,one2) ).

% and_not_num.simps(2)
tff(fact_7107_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit0,M)) = case_nat(option(num),none(num),aTP_Lamp_adq(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_7108_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = some(num,aa(num,num,bit0,M)) ).

% and_not_num.simps(7)
tff(fact_7109_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = some(num,Q2) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_7110_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_adr(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_7111_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_7112_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_7113_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_7114_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G),top_top(set(C)))),S))
       => finite673082921795544331dem_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_7115_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A2),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),A2),B2))) ) ).

% positive_rat
tff(fact_7116_Rat_Opositive__zero,axiom,
    ~ pp(aa(rat,bool,positive,zero_zero(rat))) ).

% Rat.positive_zero
tff(fact_7117_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_7118_less__rat__def,axiom,
    ! [X: rat,Y4: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y4))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y4),X))) ) ).

% less_rat_def
tff(fact_7119_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_7120_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y4 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y4 != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y4 != some(num,one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y4 != none(num) ) ) )
           => ( ! [M4: num] :
                  ( ( X = aa(num,num,bit0,M4) )
                 => ( ( Xa = one2 )
                   => ( Y4 != some(num,aa(num,num,bit0,M4)) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y4 != map_option(num,num,bit0,bit_and_not_num(M4,N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y4 != map_option(num,num,bit0,bit_and_not_num(M4,N2)) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa(num,num,bit1,M4) )
                       => ( ( Xa = one2 )
                         => ( Y4 != some(num,aa(num,num,bit0,M4)) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y4 != case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),bit_and_not_num(M4,N2)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y4 != map_option(num,num,bit0,bit_and_not_num(M4,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_7121_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit0,M),aa(num,num,bit0,N)) = map_option(num,num,bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(5)
tff(fact_7122_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = map_option(num,num,bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(9)
tff(fact_7123_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit0,M),aa(num,num,bit1,N)) = map_option(num,num,bit0,bit_and_not_num(M,N)) ).

% and_not_num.simps(6)
tff(fact_7124_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y4 )
     => ( pp(aa(product_prod(num,num),bool,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 )
             => ( ( Y4 = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,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) )
                 => ( ( Y4 = some(num,one2) )
                   => ~ pp(aa(product_prod(num,num),bool,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) )
                   => ( ( Y4 = none(num) )
                     => ~ pp(aa(product_prod(num,num),bool,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)))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ( ( Xa = one2 )
                     => ( ( Y4 = some(num,aa(num,num,bit0,M4)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),one2))) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y4 = map_option(num,num,bit0,bit_and_not_num(M4,N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa(num,num,bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y4 = map_option(num,num,bit0,bit_and_not_num(M4,N2)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ( ( Xa = one2 )
                           => ( ( Y4 = some(num,aa(num,num,bit0,M4)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),one2))) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y4 = case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),bit_and_not_num(M4,N2)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa(num,num,bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y4 = map_option(num,num,bit0,bit_and_not_num(M4,N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_7125_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_ads(product_prod(int,int),bool)) ).

% Rat.positive_def
tff(fact_7126_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,X),Xa) = Y4 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y4 != some(num,one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y4 != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y4 != some(num,one2) ) ) )
           => ( ( ? [M4: num] : X = aa(num,num,bit0,M4)
               => ( ( Xa = one2 )
                 => ( Y4 != none(num) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y4 != map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y4 != map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) ) ) )
                 => ( ( ? [M4: num] : X = aa(num,num,bit1,M4)
                     => ( ( Xa = one2 )
                       => ( Y4 != some(num,one2) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y4 != map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y4 != case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_7127_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,X),Xa) = Y4 )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y4 != none(num) ) ) )
       => ( ( ( X = one2 )
           => ! [N2: num] :
                ( ( Xa = aa(num,num,bit0,N2) )
               => ( Y4 != some(num,aa(num,num,bit1,N2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit1,N2) )
                 => ( Y4 != some(num,aa(num,num,bit0,N2)) ) ) )
           => ( ! [M4: num] :
                  ( ( X = aa(num,num,bit0,M4) )
                 => ( ( Xa = one2 )
                   => ( Y4 != some(num,aa(num,num,bit1,M4)) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y4 != map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2)) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y4 != some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa(num,num,bit1,M4) )
                       => ( ( Xa = one2 )
                         => ( Y4 != some(num,aa(num,num,bit0,M4)) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y4 != some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2))) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y4 != map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_7128_and__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),one2) = some(num,one2) ).

% and_num.simps(1)
tff(fact_7129_xor__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),one2) = none(num) ).

% xor_num.simps(1)
tff(fact_7130_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),aa(num,num,bit0,N)) = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(5)
tff(fact_7131_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),aa(num,num,bit0,N)) = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(5)
tff(fact_7132_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),one2) = some(num,one2) ).

% and_num.simps(7)
tff(fact_7133_and__num_Osimps_I3_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit1,N)) = some(num,one2) ).

% and_num.simps(3)
tff(fact_7134_and__num_Osimps_I4_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),one2) = none(num) ).

% and_num.simps(4)
tff(fact_7135_and__num_Osimps_I2_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit0,N)) = none(num) ).

% and_num.simps(2)
tff(fact_7136_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N) = some(num,Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_7137_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N) = some(num,Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_7138_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(8)
tff(fact_7139_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(6)
tff(fact_7140_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),aa(num,num,bit1,N)) = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(9)
tff(fact_7141_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit0,N)) = some(num,aa(num,num,bit1,N)) ).

% xor_num.simps(2)
tff(fact_7142_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit1,N)) = some(num,aa(num,num,bit0,N)) ).

% xor_num.simps(3)
tff(fact_7143_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),one2) = some(num,aa(num,num,bit1,M)) ).

% xor_num.simps(4)
tff(fact_7144_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),one2) = some(num,aa(num,num,bit0,M)) ).

% xor_num.simps(7)
tff(fact_7145_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_7146_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_7147_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ).

% numeral_and_num
tff(fact_7148_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ) ).

% numeral_xor_num
tff(fact_7149_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),aa(num,num,bit1,N)) = case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(9)
tff(fact_7150_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(8)
tff(fact_7151_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(6)
tff(fact_7152_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,X),Xa) = Y4 )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y4 = some(num,one2) )
               => ~ pp(aa(product_prod(num,num),bool,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) )
                 => ( ( Y4 = none(num) )
                   => ~ pp(aa(product_prod(num,num),bool,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) )
                   => ( ( Y4 = some(num,one2) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2)))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ( ( Xa = one2 )
                     => ( ( Y4 = none(num) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),one2))) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y4 = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa(num,num,bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y4 = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ( ( Xa = one2 )
                           => ( ( Y4 = some(num,one2) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),one2))) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y4 = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa(num,num,bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y4 = case_option(option(num),num,some(num,one2),aTP_Lamp_adp(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M4),N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_7153_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y4: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,X),Xa) = Y4 )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y4 = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,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) )
                 => ( ( Y4 = some(num,aa(num,num,bit1,N2)) )
                   => ~ pp(aa(product_prod(num,num),bool,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) )
                   => ( ( Y4 = some(num,aa(num,num,bit0,N2)) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2)))) ) ) )
             => ( ! [M4: num] :
                    ( ( X = aa(num,num,bit0,M4) )
                   => ( ( Xa = one2 )
                     => ( ( Y4 = some(num,aa(num,num,bit1,M4)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),one2))) ) ) )
               => ( ! [M4: num] :
                      ( ( X = aa(num,num,bit0,M4) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y4 = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M4: num] :
                        ( ( X = aa(num,num,bit0,M4) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y4 = some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2))) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M4)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M4: num] :
                          ( ( X = aa(num,num,bit1,M4) )
                         => ( ( Xa = one2 )
                           => ( ( Y4 = some(num,aa(num,num,bit0,M4)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),one2))) ) ) )
                     => ( ! [M4: num] :
                            ( ( X = aa(num,num,bit1,M4) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y4 = some(num,case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2))) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X = aa(num,num,bit1,M4) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y4 = map_option(num,num,bit0,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M4),N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M4)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_7154_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_7155_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_7156_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_7157_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_7158_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_adt(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_7159_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_adx(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),product_Pair(nat,num,N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_7160_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( zero_zero(A) = aa(rat,A,field_char_0_of_rat(A),A2) )
        <=> ( zero_zero(rat) = A2 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_7161_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_7162_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_7163_of__rat__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : aa(rat,A,field_char_0_of_rat(A),aa(num,rat,numeral_numeral(rat),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_rat_numeral_eq
tff(fact_7164_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_7165_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: 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),R2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2)) ) ) ).

% zero_less_of_rat_iff
tff(fact_7166_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R2),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_7167_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: 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),R2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R2)) ) ) ).

% zero_le_of_rat_iff
tff(fact_7168_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R2),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_7169_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R2),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_7170_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: 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),R2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R2)) ) ) ).

% one_le_of_rat_iff
tff(fact_7171_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_7172_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),aa(rat,A,field_char_0_of_rat(A),S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R2),S2)) ) ) ).

% of_rat_less_eq
tff(fact_7173_of__rat__dense,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4))
     => ? [Q4: rat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(rat,real,field_char_0_of_rat(real),Q4)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q4)),Y4)) ) ) ).

% of_rat_dense
tff(fact_7174_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_ady(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22)),aa(fun(num,A),fun(num,B),aTP_Lamp_ady(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32)),Num) ).

% num.case_distrib
tff(fact_7175_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_7176_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X2: 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,X2)) = aa(num,A,F22,X2) ).

% verit_eq_simplify(17)
tff(fact_7177_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X33: 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,X33)) = aa(num,A,F32,X33) ).

% verit_eq_simplify(18)
tff(fact_7178_of__rat__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_diff
tff(fact_7179_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_mult
tff(fact_7180_of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_divide
tff(fact_7181_nonzero__of__rat__inverse,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( A2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,inverse_inverse(rat),A2)) = aa(A,A,inverse_inverse(A),aa(rat,A,field_char_0_of_rat(A),A2)) ) ) ) ).

% nonzero_of_rat_inverse
tff(fact_7182_nonzero__of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: rat,A2: rat] :
          ( ( B2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ) ).

% nonzero_of_rat_divide
tff(fact_7183_of__rat__rat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: int,A2: int] :
          ( ( B2 != zero_zero(int) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2)) ) ) ) ).

% of_rat_rat
tff(fact_7184_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_7185_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_adz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_7186_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_aea(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_7187_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_7188_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_7189_times__rat__def,axiom,
    times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aeb(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_7190_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa),Xa))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),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,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),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_7191_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_7192_ratrel__iff,axiom,
    ! [X: product_prod(int,int),Y4: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),Y4))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y4) != 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),Y4)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y4)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).

% ratrel_iff
tff(fact_7193_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_7194_ratrel__def,axiom,
    ! [X4: 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,X4),Xa2))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X4) != 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),X4)),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),X4)) ) ) ) ).

% ratrel_def
tff(fact_7195_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_7196_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa),Xa))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),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,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_7197_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_7198_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)))
     => ( aa(set(nat),set(A),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_7199_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_aec(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_7200_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_7201_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_7202_take__upt,axiom,
    ! [I: nat,M: 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),I),M)),N))
     => ( take(nat,M,upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_7203_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),Y4: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( take(A,N,list_update(A,Xs,M,Y4)) = take(A,N,Xs) ) ) ).

% take_update_cancel
tff(fact_7204_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,N,Xs)) ) ) ).

% sorted_take
tff(fact_7205_set__take__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(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),take(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_take_subset
tff(fact_7206_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_7207_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: 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)),Ys)))
       => ( ! [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,Ys),I3) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_7208_take__bit__horner__sum__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,Bs: list(bool)] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),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)) = 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))),take(bool,N,Bs)) ) ).

% take_bit_horner_sum_bit_eq
tff(fact_7209_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_ym(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_7210_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,set(B)),A3: set(B),F3: filter(A)] :
      ( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,A3),F3)
    <=> ! [X9: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),X9))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X9),A3)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aed(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F2),A3),X9),F3) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_7211_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( ! [X8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),X8))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),A3))
           => pp(aa(set(A),bool,P,X8)) ) )
     => eventually(set(A),P,finite5375528669736107172at_top(A,A3)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_7212_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_7213_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A2: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A2),B2) = A2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A2),B2) = B2 ) ) ) ).

% ord.min_def
tff(fact_7214_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),bool),A3: set(A)] :
      ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
    <=> ? [X9: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),X9))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X9),A3))
          & ! [Y7: set(A)] :
              ( ( pp(aa(set(A),bool,finite_finite(A),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X9),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y7),A3)) )
             => pp(aa(set(A),bool,P,Y7)) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_7215_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image(set(A),filter(set(A)),aTP_Lamp_aef(set(A),fun(set(A),filter(set(A))),A3)),collect(set(A),aTP_Lamp_aeg(set(A),fun(set(A),bool),A3)))) ).

% finite_subsets_at_top_def
tff(fact_7216_card__Min__le__sum,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => 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(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image(A,nat,F2),A3)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3))) ) ).

% card_Min_le_sum
tff(fact_7217_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: 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))),M)) = 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),M),N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_7218_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_7219_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_7220_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_7221_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).

% min_top2
tff(fact_7222_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).

% min_top
tff(fact_7223_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).

% min_bot
tff(fact_7224_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_7225_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y4: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y4),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)) = Y4 ) ).

% max_min_same(4)
tff(fact_7226_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),Y4) = Y4 ) ).

% max_min_same(3)
tff(fact_7227_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),X) = X ) ).

% max_min_same(2)
tff(fact_7228_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)) = X ) ).

% max_min_same(1)
tff(fact_7229_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).

% min_Suc_Suc
tff(fact_7230_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_7231_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_7232_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),A2) ) ).

% take_bit_take_bit
tff(fact_7233_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_7234_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(1)
tff(fact_7235_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_7236_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_7237_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_7238_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_7239_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_7240_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_7241_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ) ).

% take_bit_of_mask
tff(fact_7242_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(4)
tff(fact_7243_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = 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),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(3)
tff(fact_7244_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = 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),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(2)
tff(fact_7245_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
            <=> ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_7246_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_7247_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_7248_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_7249_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_7250_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_7251_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(N,K),L)),R2) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N),K),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_7252_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y4: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y4)) ) ).

% minus_min_eq_max
tff(fact_7253_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y4: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y4)) ) ).

% minus_max_eq_min
tff(fact_7254_min__diff,axiom,
    ! [M: 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),M),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),M),N)),I) ).

% min_diff
tff(fact_7255_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ).

% min_diff_distrib_left
tff(fact_7256_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y4: 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),Y4),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_7257_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ).

% min_add_distrib_left
tff(fact_7258_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y4: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),X),Y4)) = 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),Y4)) ) ).

% of_int_min
tff(fact_7259_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_7260_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_7261_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4) = Y4 ) ) ) ).

% min_absorb2
tff(fact_7262_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4) = X ) ) ) ).

% min_absorb1
tff(fact_7263_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_7264_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: 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),Y4)),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),Y4),Z)) ) ) ) ).

% min_le_iff_disj
tff(fact_7265_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: 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),A2),B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_7266_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_7267_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_7268_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_7269_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: 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),A2),B2)),B2)) ) ).

% min.cobounded2
tff(fact_7270_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: 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),A2),B2)),A2)) ) ).

% min.cobounded1
tff(fact_7271_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_7272_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).

% min.boundedI
tff(fact_7273_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.boundedE
tff(fact_7274_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% min.orderI
tff(fact_7275_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_7276_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),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),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2))) ) ) ) ).

% min.mono
tff(fact_7277_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X4),Xa2) = X4 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X4),Xa2) = Xa2 ) ) ) ) ).

% min_def_raw
tff(fact_7278_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X)) ) ) ) ).

% Min_le
tff(fact_7279_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [Y: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_7280_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A2)) ) ) ) ).

% Min.coboundedI
tff(fact_7281_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B4)),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ) ).

% Min.subset
tff(fact_7282_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y4: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y4)) = 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),Y4)) ) ).

% of_nat_min
tff(fact_7283_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.boundedI
tff(fact_7284_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A16)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_7285_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X5)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_7286_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X))
            <=> ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_7287_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X5)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_7288_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B3)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_7289_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(N,K),L)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N),K),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_7290_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y4)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y4)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_7291_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y4)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y4)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_7292_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),P3)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_7293_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y4),P3)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_7294_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y4),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y4),P3)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_7295_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y4),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y4)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y4),P3)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_7296_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_aeh(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_7297_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_aei(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_7298_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M10: set(A),N5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M10),N5))
         => ( ( M10 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N5)),aa(set(A),A,lattic643756798350308766er_Min(A),M10))) ) ) ) ) ).

% Min_antimono
tff(fact_7299_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B4)),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.subset_imp
tff(fact_7300_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))) ) ).

% mod_exp_eq
tff(fact_7301_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),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(A,P,Xs) = some(A,aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aej(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_7302_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_ym(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_7303_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q2),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_7304_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_7305_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_7306_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_ym(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_7307_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image(A,nat,F2),A3))))) ) ).

% sum_le_card_Max
tff(fact_7308_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aTP_Lamp_kf(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_7309_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
            <=> ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_7310_inf__enat__def,axiom,
    inf_inf(extended_enat) = ord_min(extended_enat) ).

% inf_enat_def
tff(fact_7311_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A2: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A2),B2) = B2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A2),B2) = A2 ) ) ) ).

% ord.max_def
tff(fact_7312_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_7313_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max_ge
tff(fact_7314_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [Y: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_7315_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,finite_finite(A),B4))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => ? [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),B4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
                   => ? [Xa2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_7316_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max.coboundedI
tff(fact_7317_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_7318_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A16),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_7319_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_7320_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)))
            <=> ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_7321_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_7322_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B3),A2)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_7323_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M10: set(A),N5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M10),N5))
         => ( ( M10 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M10)),aa(set(A),A,lattic643756798349783984er_Max(A),N5))) ) ) ) ) ).

% Max_mono
tff(fact_7324_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B4))) ) ) ) ) ).

% Max.subset_imp
tff(fact_7325_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B4)),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ) ).

% Max.subset
tff(fact_7326_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S)))) ) ).

% card_le_Suc_Max
tff(fact_7327_Sup__nat__def,axiom,
    ! [X7: set(nat)] :
      ( ( ( X7 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X7) = zero_zero(nat) ) )
      & ( ( X7 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X7) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X7) ) ) ) ).

% Sup_nat_def
tff(fact_7328_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_aek(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_7329_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: 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),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ael(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_7330_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_7331_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_7332_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) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),aTP_Lamp_aem(nat,set(nat))),M10))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7333_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_ym(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_7334_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( linord4507533701916653071of_set(A,A3) = cons(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3),linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_7335_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,cons(A,X,Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_7336_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,cons(A,X,Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_7337_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,aa(nat,nat,suc,N),cons(A,X,Xs)) = cons(A,X,take(A,N,Xs)) ).

% take_Suc_Cons
tff(fact_7338_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),collect(int,aTP_Lamp_kv(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_7339_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,X: B,Xs: list(B)] : 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),A2),cons(B,X,Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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),A2),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_7340_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,cons(B,X,Xs)) = 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_7341_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,cons(A,X,Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_7342_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),cons(A,X,Xs)) = cons(A,X,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_7343_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,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_7344_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: 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)),Ys)))
     => ( Xs != cons(A,X,Ys) ) ) ).

% impossible_Cons
tff(fact_7345_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y4: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),cons(A,X,cons(A,Y4,Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
            & sorted_wrt(A,ord_less_eq(A),cons(A,Y4,Zs)) ) ) ) ).

% sorted2
tff(fact_7346_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: 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)),aa(list(A),set(A),set2(A),cons(A,X,Xs)))) ).

% set_subset_Cons
tff(fact_7347_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Y4: B,Ys: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y4)))
           => ( linorder_insort_key(B,A,F2,X,cons(B,Y4,Ys)) = cons(B,X,cons(B,Y4,Ys)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y4)))
           => ( linorder_insort_key(B,A,F2,X,cons(B,Y4,Ys)) = cons(B,Y4,linorder_insort_key(B,A,F2,X,Ys)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_7348_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list(nat),Q2: nat] :
      ( ( cons(nat,M,cons(nat,N,Ns)) = upt(M,Q2) )
    <=> ( cons(nat,N,Ns) = upt(aa(nat,nat,suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_7349_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = cons(A,X,replicate(A,N,X)) ).

% replicate_Suc
tff(fact_7350_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y4: A] : list_update(A,cons(A,X,Xs),zero_zero(nat),Y4) = cons(A,Y4,Xs) ).

% list_update_code(2)
tff(fact_7351_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,Y4: A] : list_update(A,cons(A,X,Xs),aa(nat,nat,suc,I),Y4) = cons(A,X,list_update(A,Xs,I,Y4)) ).

% list_update_code(3)
tff(fact_7352_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),cons(A,X,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_7353_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) )
    <=> ? [Y5: A,Ys2: list(A)] :
          ( ( Xs = cons(A,Y5,Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% length_Suc_conv
tff(fact_7354_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) )
    <=> ? [Y5: A,Ys2: list(A)] :
          ( ( Xs = cons(A,Y5,Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% Suc_length_conv
tff(fact_7355_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_aeo(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_7356_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)))
    <=> ? [X5: A,Ys2: list(A)] :
          ( ( Xs = cons(A,X5,Ys2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys2))) ) ) ).

% Suc_le_length_iff
tff(fact_7357_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),cons(A,X,Ys))
        <=> ( ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_7358_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( upt(I,J) = cons(nat,I,upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_7359_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F2: fun(B,A),A2: B] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,A2)),aa(B,A,F2,X3))) )
         => ( linorder_insort_key(B,A,F2,A2,Xs) = cons(B,A2,Xs) ) ) ) ).

% insort_is_Cons
tff(fact_7360_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N) = aa(set(int),int,lattic643756798349783984er_Max(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_aep(int,fun(int,fun(int,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_7361_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),cons(A,X21,X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_7362_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,cons(A,X,Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(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_7363_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,cons(A,X21,X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_7364_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I)),J))
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7365_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))) = cons(A,aa(nat,A,F2,zero_zero(nat)),map(nat,A,aTP_Lamp_vf(fun(nat,A),fun(nat,A),F2),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_7366_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),J))
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7367_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,cons(A,X,Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_7368_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y4: A,Xs: list(A),N: nat] :
      ( ( X != Y4 )
     => ( ( aa(nat,A,nth(A,cons(A,X,Xs)),N) = Y4 )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y4 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_7369_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y4: A] :
      ( ( cons(A,X,Xs) = replicate(A,N,Y4) )
    <=> ( ( X = Y4 )
        & 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_7370_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : aa(char,A,comm_s6883823935334413003f_char(A),char2(B0,B1,B22,B32,B42,B52,B62,B72)) = 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))),cons(bool,B0,cons(bool,B1,cons(bool,B22,cons(bool,B32,cons(bool,B42,cons(bool,B52,cons(bool,B62,cons(bool,B72,nil(bool)))))))))) ) ).

% of_char_Char
tff(fact_7371_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),cons(int,J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_7372_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_7373_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_7374_take0,axiom,
    ! [A: $tType,X4: list(A)] : take(A,zero_zero(nat),X4) = nil(A) ).

% take0
tff(fact_7375_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_7376_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_7377_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_7378_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_7379_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: 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),A2),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_7380_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(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_7381_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_7382_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = cons(list(A),nil(A),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_7383_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_7384_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y4: A] : aa(A,nat,count_list(A,nil(A)),Y4) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_7385_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_7386_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_7387_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_7388_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7389_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),cons(A,X,nil(A))) ) ).

% sorted1
tff(fact_7390_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( take(A,N,cons(A,X,Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( take(A,N,cons(A,X,Xs)) = 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_7391_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A3) = nil(B) )
          <=> ( A3 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7392_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,cons(A,X,nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_7393_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,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_7394_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7395_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_7396_take__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : take(A,N,append(A,Xs,Ys)) = 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)),Ys)) ).

% take_append
tff(fact_7397_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)),cons(nat,K,nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_7398_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)),cons(nat,aa(nat,nat,suc,K),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_7399_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = cons(nat,aa(num,nat,numeral_numeral(nat),M),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_7400_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),cons(nat,J,nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_7401_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),cons(nat,J,nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7402_list__encode_Ocases,axiom,
    ! [X: list(nat)] :
      ( ( X != nil(nat) )
     => ~ ! [X3: nat,Xs2: list(nat)] : X != cons(nat,X3,Xs2) ) ).

% list_encode.cases
tff(fact_7403_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_7404_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7405_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa4)) ) ) ) ) ) ).

% sorted_append
tff(fact_7406_inj__list__encode,axiom,
    ! [A3: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A3) ).

% inj_list_encode
tff(fact_7407_list__encode__eq,axiom,
    ! [X: list(nat),Y4: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = aa(list(nat),nat,nat_list_encode,Y4) )
    <=> ( X = Y4 ) ) ).

% list_encode_eq
tff(fact_7408_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_7409_surj__list__encode,axiom,
    aa(set(list(nat)),set(nat),image(list(nat),nat,nat_list_encode),top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_7410_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_7411_list__encode_Oelims,axiom,
    ! [X: list(nat),Y4: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y4 )
     => ( ( ( X = nil(nat) )
         => ( Y4 != zero_zero(nat) ) )
       => ~ ! [X3: nat,Xs2: list(nat)] :
              ( ( X = cons(nat,X3,Xs2) )
             => ( Y4 != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7412_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) )
    <=> ? [Y5: A,Ys2: list(A)] :
          ( ( Xs = append(A,Ys2,cons(A,Y5,nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_7413_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,cons(A,X,nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_7414_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: 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,Ys)),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,Ys)),N) = aa(nat,A,nth(A,Ys),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_7415_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: 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,Ys),N,X) = append(A,list_update(A,Xs,N,X),Ys) ) )
      & ( ~ 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,Ys),N,X) = append(A,Xs,list_update(A,Ys,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_7416_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
       => ( upt(I,J) = cons(nat,I,upt(aa(nat,nat,suc,I),J)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
       => ( upt(I,J) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_7417_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),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),A2),Ys))) ) ).

% horner_sum_append
tff(fact_7418_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
           => ( linorder_insort_key(A,A,aTP_Lamp_nb(A,A),A2,Xs) = append(A,Xs,cons(A,A2,nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_7419_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),cons(A,aa(nat,A,nth(A,Xs),I),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_7420_list__encode_Opelims,axiom,
    ! [X: list(nat),Y4: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y4 )
     => ( pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),X))
       => ( ( ( X = nil(nat) )
           => ( ( Y4 = zero_zero(nat) )
             => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),nil(nat))) ) )
         => ~ ! [X3: nat,Xs2: list(nat)] :
                ( ( X = cons(nat,X3,Xs2) )
               => ( ( Y4 = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),cons(nat,X3,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7421_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),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)))
       => ( ( M != N )
         => ( aa(nat,A,nth(A,append(A,take(A,N,Xs),append(A,cons(A,X,nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_7422_pos__n__replace,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y4: 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,cons(A,Y4,nil(A)),drop(A,aa(nat,nat,suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_7423_drop0,axiom,
    ! [A: $tType,X4: list(A)] : drop(A,zero_zero(nat),X4) = X4 ).

% drop0
tff(fact_7424_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),cons(A,X,Xs)) = drop(A,N,Xs) ).

% drop_Suc_Cons
tff(fact_7425_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_7426_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_7427_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_7428_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_7429_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_7430_drop__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : drop(A,N,append(A,Xs,Ys)) = 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)),Ys)) ).

% drop_append
tff(fact_7431_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),cons(A,X,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_7432_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_7433_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7434_drop__take,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,take(A,M,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),drop(A,N,Xs)) ).

% drop_take
tff(fact_7435_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_7436_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,N,Xs)) ) ) ).

% sorted_drop
tff(fact_7437_set__drop__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(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),drop(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_drop_subset
tff(fact_7438_drop__update__swap,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( drop(A,M,list_update(A,Xs,N,X)) = list_update(A,drop(A,M,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M),X) ) ) ).

% drop_update_swap
tff(fact_7439_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( drop(A,N,cons(A,X,Xs)) = cons(A,X,Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( drop(A,N,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_7440_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_7441_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)))
     => ( 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_7442_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_7443_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),cons(A,aa(nat,A,nth(A,Xs),I),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_7444_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: 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,A2) = append(A,take(A,I,Xs),cons(A,A2,drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_7445_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_aer(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_7446_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y4: list(int)] :
      ( ( upto(X,Xa) = Y4 )
     => ( pp(aa(product_prod(int,int),bool,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))
               => ( Y4 = 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))
               => ( Y4 = nil(int) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,X),Xa))) ) ) ) ).

% upto.pelims
tff(fact_7447_nth__upto,axiom,
    ! [I: int,K: nat,J: 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))),J))
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_7448_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I)),one_one(int))) ).

% length_upto
tff(fact_7449_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = cons(int,aa(num,int,numeral_numeral(int),M),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_7450_upto__rec__numeral_I4_J,axiom,
    ! [M: 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),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_7451_upto__rec__numeral_I3_J,axiom,
    ! [M: 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),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_7452_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = cons(int,aa(num,int,numeral_numeral(int),M),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_7453_sorted__upto,axiom,
    ! [M: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N)) ).

% sorted_upto
tff(fact_7454_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = append(int,upto(I,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_7455_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_7456_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_7457_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
       => ( upto(I,J) = cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
       => ( upto(I,J) = nil(int) ) ) ) ).

% upto.simps
tff(fact_7458_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y4: list(int)] :
      ( ( upto(X,Xa) = Y4 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
         => ( Y4 = 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))
         => ( Y4 = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_7459_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( upto(I,J) = cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_7460_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( upto(I,J) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,nil(int))) ) ) ).

% upto_rec2
tff(fact_7461_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_7462_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_7463_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_aes(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_7464_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I),J)))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_7465_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_7466_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),cons(A,hd(A,drop(A,N,Xs)),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_7467_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( hd(A,replicate(A,N,X)) = X ) ) ).

% hd_replicate
tff(fact_7468_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
     => ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_7469_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_7470_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_7471_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_7472_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_7473_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_7474_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_7475_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y4: 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,Y4)) = 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)),Y4) ) ) ).

% rev_update
tff(fact_7476_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_7477_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_7478_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_7479_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),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),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_7480_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y4: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y4) = Y4 ) )
            & ( ( Xs != nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y4) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y4) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_7481_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( 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),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_aet(nat,fun(list(A),bool),I),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_7482_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_aeu(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_aet(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_7483_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,P,Xs))) ) ) ).

% sorted_filter
tff(fact_7484_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_aev(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F2),G),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_7485_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_aew(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs),Xs)) ) ).

% sorted_same
tff(fact_7486_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(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),filter2(A,P,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_7487_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_7488_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_aex(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P),Xs)) ) ).

% sum_list_map_filter'
tff(fact_7489_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,filter2(A,P,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs)))) ).

% sum_list_filter_le_nat
tff(fact_7490_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P: fun(B,bool),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P,X3))
               => ( aa(B,A,F2,X3) = 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_7491_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => ( pp(aa(B,bool,P,X))
           => ( filter2(B,P,linorder_insort_key(B,A,F2,X,Xs)) = linorder_insort_key(B,A,F2,X,filter2(B,P,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_7492_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y4: 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,Y4),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_aey(A,fun(A,bool),Y4),Xs)) ).

% set_minus_filter_out
tff(fact_7493_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A2: B,Xs: list(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
           => ( ( hd(B,filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aez(B,fun(fun(B,A),fun(B,bool)),A2),F2),Xs)) = A2 )
             => ( linorder_insort_key(B,A,F2,A2,remove1(B,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7494_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_afa(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_afb(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_afc(list(B),bool),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7495_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_aes(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_afd(list(A),bool),Xs)) ).

% transpose_max_length
tff(fact_7496_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_aet(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_7497_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_afd(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_7498_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_afe(fun(B,A),fun(A,fun(B,bool)),F2),T2),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_afe(fun(B,A),fun(A,fun(B,bool)),F2),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7499_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_7500_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_7501_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
         => 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7502_nths__shift__lemma,axiom,
    ! [A: $tType,A3: set(nat),Xs: list(A),I: nat] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_aff(set(nat),fun(product_prod(A,nat),bool),A3),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_afg(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A3),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7503_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A3: set(nat)] : nths(A,cons(A,X,L),A3) = append(A,if(list(A),aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3),cons(A,X,nil(A)),nil(A)),nths(A,L,collect(nat,aTP_Lamp_afh(set(nat),fun(nat,bool),A3)))) ).

% nths_Cons
tff(fact_7504_nths__singleton,axiom,
    ! [A: $tType,A3: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,cons(A,X,nil(A)),A3) = cons(A,X,nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,cons(A,X,nil(A)),A3) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_7505_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs))) ).

% set_nths_subset
tff(fact_7506_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_7507_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,collect(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_7508_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat)] : nths(A,Xs,A3) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_aff(set(nat),fun(product_prod(A,nat),bool),A3),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7509_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_afi(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_afj(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_7510_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_7511_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_7512_tl__upt,axiom,
    ! [M: nat,N: nat] : tl(nat,upt(M,N)) = upt(aa(nat,nat,suc,M),N) ).

% tl_upt
tff(fact_7513_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_7514_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_7515_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_7516_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_7517_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_7518_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_7519_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),tl(A,Xs)) ) ) ).

% sorted_tl
tff(fact_7520_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_7521_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_7522_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_7523_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_7524_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)) = cons(A,X,nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_7525_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_7526_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,N),Xs) = cons(A,hd(A,Xs),take(A,N,tl(A,Xs))) ) ) ).

% take_Suc
tff(fact_7527_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F7: fun(nat,nat)] :
          ( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F7))
          & ( aa(set(nat),set(nat),image(nat,nat,F7),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [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,Ys),aa(nat,nat,F7,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,F7,I2) = aa(nat,nat,F7,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_7528_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_afk(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_7529_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
        <=> ( algebr8660921524188924756oprime(A,A2,C2)
            & algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_7530_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( algebr8660921524188924756oprime(A,C2,A2)
            & algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_7531_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( algebr8660921524188924756oprime(int,aa(int,int,abs_abs(int),K),L)
    <=> algebr8660921524188924756oprime(int,K,L) ) ).

% coprime_abs_left_iff
tff(fact_7532_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( algebr8660921524188924756oprime(int,K,aa(int,int,abs_abs(int),L))
    <=> algebr8660921524188924756oprime(int,K,L) ) ).

% coprime_abs_right_iff
tff(fact_7533_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,A2)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A))) ) ) ).

% coprime_self
tff(fact_7534_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A2,modulo_modulo(A,B2,A2))
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_7535_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A2,B2),B2)
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_7536_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,N: nat] :
          ( algebr8660921524188924756oprime(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_7537_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,N: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N),B2)
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_7538_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,zero_zero(A))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A))) ) ) ).

% coprime_0_right_iff
tff(fact_7539_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A2)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A))) ) ) ).

% coprime_0_left_iff
tff(fact_7540_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_7541_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_7542_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: 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),A2),B2)),one_one(A)))
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_gcd
tff(fact_7543_coprime__right__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% coprime_right_2_iff_odd
tff(fact_7544_coprime__left__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% coprime_left_2_iff_odd
tff(fact_7545_normalize__stable,axiom,
    ! [Q2: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2))
     => ( algebr8660921524188924756oprime(int,P3,Q2)
       => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) ) ) ) ).

% normalize_stable
tff(fact_7546_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ).

% mono_Sup
tff(fact_7547_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5))))) ) ) ).

% mono_SUP
tff(fact_7548_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,F2,P3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),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_7549_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( ( A2 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ).

% div_gcd_coprime
tff(fact_7550_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,P3)),P3))
           => 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))),P3)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_7551_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A3: A,B4: A,N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B4))
           => 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),A3)),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),B4))) ) ) ) ).

% funpow_mono
tff(fact_7552_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y4))) ) ) ) ).

% monoD
tff(fact_7553_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y4))) ) ) ) ).

% monoE
tff(fact_7554_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))) )
         => pp(aa(fun(A,B),bool,order_mono(A,B),F2)) ) ) ).

% monoI
tff(fact_7555_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
        <=> ! [X5: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X5)),aa(A,B,F2,Y5))) ) ) ) ).

% mono_def
tff(fact_7556_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,I)),aa(nat,A,A3,aa(nat,nat,suc,I)))) ) ) ).

% incseq_SucD
tff(fact_7557_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N2)),aa(nat,A,X7,aa(nat,nat,suc,N2))))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),X7)) ) ) ).

% incseq_SucI
tff(fact_7558_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F2))
        <=> ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4)))) ) ) ).

% incseq_Suc_iff
tff(fact_7559_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,I)),aa(nat,A,F2,J))) ) ) ) ).

% incseqD
tff(fact_7560_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X7: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X7))
        <=> ! [M5: nat,N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,M5)),aa(nat,A,X7,N4))) ) ) ) ).

% incseq_def
tff(fact_7561_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( pp(aa(fun(A,B),bool,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,Y4)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ).

% mono_invE
tff(fact_7562_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y4: A] :
          ( pp(aa(fun(A,B),bool,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,Y4)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4)) ) ) ) ).

% mono_strict_invE
tff(fact_7563_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
           => ( algebr8660921524188924756oprime(A,A2,B2)
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)) ) ) ) ) ).

% divides_mult
tff(fact_7564_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_7565_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),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),A2),B2)) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_7566_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ! [C4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A2))
             => ( 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))) ) )
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprimeI
tff(fact_7567_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
        <=> ! [C6: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C6),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C6),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C6),one_one(A))) ) ) ) ) ).

% coprime_def
tff(fact_7568_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A2,B2)
         => ~ ! [C4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A2))
               => ( 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))) ) ) ) ) ).

% not_coprimeE
tff(fact_7569_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
         => ( 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,A2,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_7570_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y4))
         => ( algebr8660921524188924756oprime(A,X,Y4)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A))) ) ) ) ).

% coprime_absorb_left
tff(fact_7571_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D2: A,A2: 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),A2))
                 => ( 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),A2))
                   => ( 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,A2,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_7572_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y4),X))
         => ( algebr8660921524188924756oprime(A,X,Y4)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y4),one_one(A))) ) ) ) ).

% coprime_absorb_right
tff(fact_7573_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A2))
           => ( 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_7574_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_7575_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_7576_mono__Suc,axiom,
    pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).

% mono_Suc
tff(fact_7577_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,A2)
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_commute
tff(fact_7578_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D2))
           => ( algebr8660921524188924756oprime(A,C2,D2)
             => algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_7579_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A] : pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A2))) ) ).

% mono_add
tff(fact_7580_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,one_one(A)) ) ).

% coprime_1_right
tff(fact_7581_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,one_one(A),A2) ) ).

% coprime_1_left
tff(fact_7582_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),A2) ) ).

% coprime_diff_one_left
tff(fact_7583_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_7584_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_7585_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_7586_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_7587_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_7588_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A2: A,N: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),N),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A2,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_7589_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N: A,A2: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A2),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B2),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A2,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_7590_coprime__crossproduct__int,axiom,
    ! [A2: int,D2: int,B2: int,C2: int] :
      ( algebr8660921524188924756oprime(int,A2,D2)
     => ( algebr8660921524188924756oprime(int,B2,C2)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D2)) )
        <=> ( ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C2) = aa(int,int,abs_abs(int),D2) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_7591_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => pp(aa(fun(A,A),bool,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_7592_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_afm(fun(A,A),fun(nat,A),Q))) ) ) ).

% mono_funpow
tff(fact_7593_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N))) ) ).

% mono_times_nat
tff(fact_7594_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A2))) ) ) ).

% mono_mult
tff(fact_7595_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J: nat,X: A,Y4: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4))
             => ( 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)),J),F2),Y4))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_7596_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A3: A,B4: A] :
          ( pp(aa(fun(A,B),bool,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,A3)),aa(A,B,F2,B4))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4)))) ) ) ).

% mono_sup
tff(fact_7597_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ? [A17: A,B6: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A17),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),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),A2),B2)) )
              & algebr8660921524188924756oprime(A,A17,B6) ) ) ) ).

% gcd_coprime_exists
tff(fact_7598_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,A5: A,B5: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A5),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B5),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
             => algebr8660921524188924756oprime(A,A5,B5) ) ) ) ) ).

% gcd_coprime
tff(fact_7599_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A2,C2) ) ) ).

% invertible_coprime
tff(fact_7600_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => pp(aa(fun(A,B),bool,order_mono(A,B),F2)) ) ) ).

% strict_mono_mono
tff(fact_7601_coprime__common__divisor__int,axiom,
    ! [A2: int,B2: int,X: int] :
      ( algebr8660921524188924756oprime(int,A2,B2)
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),A2))
       => ( 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_7602_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A4: int,B3: int] :
          ( ( Q2 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B3) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
           => ~ algebr8660921524188924756oprime(int,A4,B3) ) ) ).

% Rat_cases
tff(fact_7603_Rat__induct,axiom,
    ! [P: fun(rat,bool),Q2: rat] :
      ( ! [A4: int,B3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
         => ( algebr8660921524188924756oprime(int,A4,B3)
           => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B3))) ) )
     => pp(aa(rat,bool,P,Q2)) ) ).

% Rat_induct
tff(fact_7604_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A3: A,B4: A] :
          ( pp(aa(fun(A,B),bool,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),A3),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(A,B,F2,B4)))) ) ) ).

% mono_inf
tff(fact_7605_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3)))) ) ) ).

% mono_Inf
tff(fact_7606_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5)))) ) ) ).

% mono_INF
tff(fact_7607_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X7: fun(nat,A),L6: A,N: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X7))
         => ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X7,N)),L6)) ) ) ) ).

% incseq_le
tff(fact_7608_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,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)),M),F2),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_7609_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,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)),M),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_7610_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A4: int,B3: int] :
          ( ( Q2 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B3) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
           => ( ( A4 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A4,B3) ) ) )
     => ( Q2 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_7611_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)))
         => ~ ! [A4: int,B3: int] :
                ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
               => ( algebr8660921524188924756oprime(int,A4,B3)
                 => ( X != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A4)),aa(int,A,ring_1_of_int(A),B3)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_7612_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod(int,int)] :
      ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),X3)) )
      & 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),X3)))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X3),aa(product_prod(int,int),int,product_snd(int,int),X3))
      & ! [Y3: product_prod(int,int)] :
          ( ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y3)),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
            & 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),Y3)))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y3),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
         => ( Y3 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_7613_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))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_afn(nat,fun(nat,nat),K))) ) ).

% mono_ge2_power_minus_self
tff(fact_7614_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite(A),aa(set(nat),set(A),image(nat,A,F2),top_top(set(nat)))))
         => ( pp(aa(fun(nat,A),bool,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] :
                  ( ! [N7: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N9))
                     => ! [M2: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N9))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N7))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,M2)),aa(nat,A,F2,N7))) ) ) )
                  & ! [N7: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N7))
                     => ( aa(nat,A,F2,N9) = aa(nat,A,F2,N7) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_7615_tendsto__at__topI__sequentially__real,axiom,
    ! [F2: fun(real,real),Y4: real] :
      ( pp(aa(fun(real,real),bool,order_mono(real,real),F2))
     => ( filterlim(nat,real,aTP_Lamp_afo(fun(real,real),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Y4),at_top(nat))
       => filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Y4),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially_real
tff(fact_7616_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X7: set(A)] :
      ( countable_countable(A,X7)
     => ~ ! [F4: 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),F4,I4)),X7))
           => ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F4,I4)),aa(nat,set(A),F4,aa(nat,nat,suc,I4))))
             => ( ! [I4: nat] : pp(aa(set(A),bool,finite_finite(A),aa(nat,set(A),F4,I4)))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F4),top_top(set(nat)))) != X7 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7617_countable__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( countable_countable(A,A3)
     => countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ).

% countable_Diff
tff(fact_7618_coprime__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))
    <=> algebr8660921524188924756oprime(nat,M,N) ) ).

% coprime_int_iff
tff(fact_7619_countable__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
    <=> countable_countable(A,A3) ) ).

% countable_Diff_eq
tff(fact_7620_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_7621_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_7622_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_7623_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_7624_coprime__crossproduct__nat,axiom,
    ! [A2: nat,D2: nat,B2: nat,C2: nat] :
      ( algebr8660921524188924756oprime(nat,A2,D2)
     => ( algebr8660921524188924756oprime(nat,B2,C2)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D2) )
        <=> ( ( A2 = B2 )
            & ( C2 = D2 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_7625_coprime__Suc__left__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).

% coprime_Suc_left_nat
tff(fact_7626_coprime__Suc__right__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).

% coprime_Suc_right_nat
tff(fact_7627_coprime__common__divisor__nat,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( algebr8660921524188924756oprime(nat,A2,B2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),A2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),B2))
         => ( X = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_7628_mono__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B4: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F2))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B4))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))) ) ).

% mono_Un
tff(fact_7629_incseq__bounded,axiom,
    ! [X7: fun(nat,real),B4: real] :
      ( pp(aa(fun(nat,real),bool,order_mono(nat,real),X7))
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X7,I3)),B4))
       => bfun(nat,real,X7,at_top(nat)) ) ) ).

% incseq_bounded
tff(fact_7630_mono__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B4: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F2))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B4)))) ) ).

% mono_Int
tff(fact_7631_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ) ).

% ccInf_greatest
tff(fact_7632_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
          <=> ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X5)) ) ) ) ) ).

% le_ccInf_iff
tff(fact_7633_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V)) ) ) ) ) ).

% ccInf_lower2
tff(fact_7634_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ) ).

% ccInf_lower
tff(fact_7635_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( countable_countable(A,B4)
         => ( countable_countable(A,A3)
           => ( ! [B3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),B4))
                 => ? [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B3)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ) ) ).

% ccInf_mono
tff(fact_7636_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ) ).

% ccInf_superset_mono
tff(fact_7637_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( countable_countable(A,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ) ).

% ccSup_subset_mono
tff(fact_7638_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( countable_countable(A,B4)
         => ( countable_countable(A,A3)
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ? [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ) ) ).

% ccSup_mono
tff(fact_7639_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ) ).

% ccSup_least
tff(fact_7640_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% ccSup_upper
tff(fact_7641_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
          <=> ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B2)) ) ) ) ) ).

% ccSup_le_iff
tff(fact_7642_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% ccSup_upper2
tff(fact_7643_uncountable__minus__countable,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ~ countable_countable(A,A3)
     => ( countable_countable(A,B4)
       => ~ countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ) ).

% uncountable_minus_countable
tff(fact_7644_infinite__countable__subset_H,axiom,
    ! [A: $tType,X7: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),X7))
     => ? [C8: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C8),X7))
          & countable_countable(A,C8)
          & ~ pp(aa(set(A),bool,finite_finite(A),C8)) ) ) ).

% infinite_countable_subset'
tff(fact_7645_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T6: set(A)] :
      ( countable_countable(A,T6)
     => countable_countable(set(A),collect(set(A),aTP_Lamp_aeg(set(A),fun(set(A),bool),T6))) ) ).

% countable_Collect_finite_subset
tff(fact_7646_countable__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( countable_countable(A,B4)
       => countable_countable(A,A3) ) ) ).

% countable_subset
tff(fact_7647_all__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S))) )
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
            & inj_on(B,A,F2,T9) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% all_countable_subset_image_inj
tff(fact_7648_ex__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
          & inj_on(B,A,F2,T9)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% ex_countable_subset_image_inj
tff(fact_7649_countable__image__eq__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image(B,A,F2),S))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
          & ( aa(set(B),set(A),image(B,A,F2),S) = aa(set(B),set(A),image(B,A,F2),T9) )
          & inj_on(B,A,F2,T9) ) ) ).

% countable_image_eq_inj
tff(fact_7650_all__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S))) )
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% all_countable_subset_image
tff(fact_7651_ex__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F2),S)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T9))) ) ) ).

% ex_countable_subset_image
tff(fact_7652_countable__subset__image,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( ( countable_countable(A,B4)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),A3))) )
    <=> ? [A18: set(B)] :
          ( countable_countable(B,A18)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A18),A3))
          & ( B4 = aa(set(B),set(A),image(B,A,F2),A18) ) ) ) ).

% countable_subset_image
tff(fact_7653_countable__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image(B,A,F2),S))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S))
          & ( aa(set(B),set(A),image(B,A,F2),S) = aa(set(B),set(A),image(B,A,F2),T9) ) ) ) ).

% countable_image_eq
tff(fact_7654_ccSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_7655_ccSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U))
          <=> ! [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X5)),U)) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_7656_ccSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ) ).

% ccSUP_upper
tff(fact_7657_ccSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),U)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U)) ) ) ) ).

% ccSUP_least
tff(fact_7658_ccSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B4: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B4)
           => ( ! [N2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
                 => ? [X4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),B4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,N2)),aa(C,A,G,X4))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B4)))) ) ) ) ) ).

% ccSUP_mono
tff(fact_7659_ccINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))) ) ) ) ).

% ccINF_greatest
tff(fact_7660_le__ccINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))))
          <=> ! [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X5))) ) ) ) ) ).

% le_ccINF_iff
tff(fact_7661_ccINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),U)) ) ) ) ) ).

% ccINF_lower2
tff(fact_7662_ccINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(B,A,F2,I))) ) ) ) ).

% ccINF_lower
tff(fact_7663_ccINF__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B4: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B4)
           => ( ! [M4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),M4),B4))
                 => ? [X4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(C,A,G,M4))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,G),B4)))) ) ) ) ) ).

% ccINF_mono
tff(fact_7664_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_7665_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_7666_incseq__convergent,axiom,
    ! [X7: fun(nat,real),B4: real] :
      ( pp(aa(fun(nat,real),bool,order_mono(nat,real),X7))
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X7,I3)),B4))
       => ~ ! [L7: real] :
              ( filterlim(nat,real,X7,topolo7230453075368039082e_nhds(real,L7),at_top(nat))
             => ~ ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X7,I4)),L7)) ) ) ) ).

% incseq_convergent
tff(fact_7667_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B4)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_7668_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B4)
           => 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(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_7669_ccSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,B4)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),B4)))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_7670_ccINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B4)))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_7671_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ) ).

% mono_ccSup
tff(fact_7672_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5))))) ) ) ) ).

% mono_ccSUP
tff(fact_7673_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F2: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5)))) ) ) ) ).

% mono_ccINF
tff(fact_7674_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3)))) ) ) ) ).

% mono_ccInf
tff(fact_7675_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)))
     => ~ ! [M4: 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),M4)),aa(nat,real,semiring_1_of_nat(real),N2)) )
             => ~ algebr8660921524188924756oprime(nat,M4,N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_7676_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3)))
     => ( pp(aa(fun(nat,real),bool,order_mono(nat,real),F2))
       => ( order_strict_mono(nat,nat,G)
         => ( bfun(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_afq(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),F2),G),at_top(nat))
          <=> bfun(nat,real,F2,at_top(nat)) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_7677_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_ads(product_prod(int,int),bool)),aTP_Lamp_ads(product_prod(int,int),bool))) ).

% Rat.positive.rsp
tff(fact_7678_vanishes__mult__bounded,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( ? [A16: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A16))
          & ! [N2: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X7,N2))),A16)) )
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6))) ) ) ).

% vanishes_mult_bounded
tff(fact_7679_vanishes__const,axiom,
    ! [C2: rat] :
      ( pp(vanishes(aTP_Lamp_afs(rat,fun(nat,rat),C2)))
    <=> ( C2 = zero_zero(rat) ) ) ).

% vanishes_const
tff(fact_7680_vanishes__diff,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6))) ) ) ).

% vanishes_diff
tff(fact_7681_vanishes__add,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( pp(vanishes(X7))
     => ( pp(vanishes(Y6))
       => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6))) ) ) ).

% vanishes_add
tff(fact_7682_vanishes__minus,axiom,
    ! [X7: fun(nat,rat)] :
      ( pp(vanishes(X7))
     => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_afv(fun(nat,rat),fun(nat,rat)),X7))) ) ).

% vanishes_minus
tff(fact_7683_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R,bNF_rel_fun(nat,nat,A,B,fequal(nat),R)),power_power(A)),power_power(B))) ) ) ) ).

% power_transfer
tff(fact_7684_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7685_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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,R,R),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),R),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7686_rel__fun__Collect__case__prodD,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,A3: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool)),F2: fun(A,C),G: fun(B,D),X7: set(product_prod(A,B)),X: product_prod(A,B)] :
      ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A3,B4),F2),G))
     => ( 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))),X7),collect(product_prod(A,B),product_case_prod(A,B,bool,A3))))
       => ( 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)),X),X7))
         => pp(aa(D,bool,aa(C,fun(D,bool),B4,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F2),product_fst(A,B)),X)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),comp(B,D,product_prod(A,B),G),product_snd(A,B)),X))) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_7687_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_adt(product_prod(int,int),A)),aTP_Lamp_adt(product_prod(int,int),A))) ) ).

% of_rat.rsp
tff(fact_7688_vanishes__def,axiom,
    ! [X7: fun(nat,rat)] :
      ( pp(vanishes(X7))
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X7,N4))),R5)) ) ) ) ).

% vanishes_def
tff(fact_7689_vanishesI,axiom,
    ! [X7: 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,X7,N2))),R3)) ) )
     => pp(vanishes(X7)) ) ).

% vanishesI
tff(fact_7690_vanishesD,axiom,
    ! [X7: fun(nat,rat),R2: rat] :
      ( pp(vanishes(X7))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
       => ? [K2: nat] :
          ! [N7: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N7))
           => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X7,N7))),R2)) ) ) ) ).

% vanishesD
tff(fact_7691_fun_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,R: fun(A,fun(B,bool)),A2: fun(D,A),B2: fun(D,B)] :
      ( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R),A2),B2))
    <=> ? [Z5: fun(D,product_prod(A,B))] :
          ( pp(aa(set(fun(D,product_prod(A,B))),bool,aa(fun(D,product_prod(A,B)),fun(set(fun(D,product_prod(A,B))),bool),member(fun(D,product_prod(A,B))),Z5),collect(fun(D,product_prod(A,B)),aTP_Lamp_afw(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R))))
          & ( aa(fun(D,product_prod(A,B)),fun(D,A),comp(product_prod(A,B),A,D,product_fst(A,B)),Z5) = A2 )
          & ( aa(fun(D,product_prod(A,B)),fun(D,B),comp(product_prod(A,B),B,D,product_snd(A,B)),Z5) = B2 ) ) ) ).

% fun.in_rel
tff(fact_7692_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7693_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_afx(int,fun(int,product_prod(int,int)))),aTP_Lamp_afx(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_7694_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S: 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))),S,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),S),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S),bNF_rel_fun(num,num,A,B,fequal(num),S)))),case_num(A)),case_num(B))) ).

% num.case_transfer
tff(fact_7695_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_7696_less__eq__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_eq(int)),ord_less_eq(int))) ).

% less_eq_integer.rsp
tff(fact_7697_less__eq__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_eq(nat)),ord_less_eq(nat))) ).

% less_eq_natural.rsp
tff(fact_7698_times__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))),times_times(nat)),times_times(nat))) ).

% times_natural.rsp
tff(fact_7699_times__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))),times_times(int)),times_times(int))) ).

% times_integer.rsp
tff(fact_7700_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_7701_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_7702_set__bit__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))),bit_se5668285175392031749et_bit(nat)),bit_se5668285175392031749et_bit(nat))) ).

% set_bit_natural.rsp
tff(fact_7703_set__bit__integer_Orsp,axiom,
    pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(int))) ).

% set_bit_integer.rsp
tff(fact_7704_unset__bit__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))),bit_se2638667681897837118et_bit(nat)),bit_se2638667681897837118et_bit(nat))) ).

% unset_bit_natural.rsp
tff(fact_7705_unset__bit__integer_Orsp,axiom,
    pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(int))) ).

% unset_bit_integer.rsp
tff(fact_7706_flip__bit__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))),bit_se8732182000553998342ip_bit(nat)),bit_se8732182000553998342ip_bit(nat))) ).

% flip_bit_natural.rsp
tff(fact_7707_flip__bit__integer_Orsp,axiom,
    pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(int))) ).

% flip_bit_integer.rsp
tff(fact_7708_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_7709_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_7710_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_7711_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_afy(num,fun(num,int))),aTP_Lamp_afy(num,fun(num,int)))) ).

% sub.rsp
tff(fact_7712_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,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),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_7713_times__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aeb(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aeb(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_7714_plus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_adz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_adz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_7715_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),P: fun(A,fun(C,bool)),Q: fun(B,fun(D,bool)),A2: product_prod(A,B),A3: set(product_prod(A,B)),B2: product_prod(C,D),B4: set(product_prod(C,D))] :
      ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),bool),bNF_rel_fun(A,B,fun(C,bool),fun(D,bool),R1,bNF_rel_fun(C,D,bool,bool,R22,fequal(bool))),P),Q))
     => ( 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)),A2),A3))
       => ( pp(aa(set(product_prod(C,D)),bool,aa(product_prod(C,D),fun(set(product_prod(C,D)),bool),member(product_prod(C,D)),B2),B4))
         => ( 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))),A3),collect(product_prod(A,B),product_case_prod(A,B,bool,R1))))
           => ( pp(aa(set(product_prod(C,D)),bool,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),bool),ord_less_eq(set(product_prod(C,D))),B4),collect(product_prod(C,D),product_case_prod(C,D,bool,R22))))
             => ( pp(aa(C,bool,aa(A,fun(C,bool),P,aa(product_prod(A,B),A,product_fst(A,B),A2)),aa(product_prod(C,D),C,product_fst(C,D),B2)))
              <=> pp(aa(D,bool,aa(B,fun(D,bool),Q,aa(product_prod(A,B),B,product_snd(A,B),A2)),aa(product_prod(C,D),D,product_snd(C,D),B2))) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_7716_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_aea(product_prod(int,int),product_prod(int,int))),aTP_Lamp_aea(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_7717_fun_Orel__mono,axiom,
    ! [D: $tType,B: $tType,A: $tType,R: 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))),R),Ra))
     => pp(aa(fun(fun(D,A),fun(fun(D,B),bool)),bool,aa(fun(fun(D,A),fun(fun(D,B),bool)),fun(fun(fun(D,A),fun(fun(D,B),bool)),bool),ord_less_eq(fun(fun(D,A),fun(fun(D,B),bool))),bNF_rel_fun(D,D,A,B,fequal(D),R)),bNF_rel_fun(D,D,A,B,fequal(D),Ra))) ) ).

% fun.rel_mono
tff(fact_7718_plus__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_adz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_7719_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_aea(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_7720_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_7721_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_afx(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_7722_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_adt(product_prod(int,int),A)),field_char_0_of_rat(A))) ) ).

% of_rat.transfer
tff(fact_7723_times__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aeb(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_7724_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_ads(product_prod(int,int),bool)),positive)) ).

% Rat.positive.transfer
tff(fact_7725_fun__mono,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,C3: fun(A,fun(B,bool)),A3: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool)),D4: fun(C,fun(D,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))),C3),A3))
     => ( pp(aa(fun(C,fun(D,bool)),bool,aa(fun(C,fun(D,bool)),fun(fun(C,fun(D,bool)),bool),ord_less_eq(fun(C,fun(D,bool))),B4),D4))
       => pp(aa(fun(fun(A,C),fun(fun(B,D),bool)),bool,aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(A,C),fun(fun(B,D),bool))),bNF_rel_fun(A,B,C,D,A3,B4)),bNF_rel_fun(A,B,C,D,C3,D4))) ) ) ).

% fun_mono
tff(fact_7726_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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_7727_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_7728_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_afz(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_7729_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_nf(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_7730_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_7731_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_7732_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_ng(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).

% of_int.transfer
tff(fact_7733_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_ni(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_7734_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_nk(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_7735_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_nm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_7736_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_no(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_7737_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S: 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))),S,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,S,S)),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,S,S)),bNF_rel_fun(num,num,A,B,fequal(num),S)))),rec_num(A)),rec_num(B))) ).

% num.rec_transfer
tff(fact_7738_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),A3: set(A),G: fun(B,nat),B4: set(B)] :
      ( inj_on(A,nat,F2,A3)
     => ( inj_on(B,nat,G,B4)
       => ( ! [A4: A,B3: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),B4))
               => algebr8660921524188924756oprime(nat,aa(A,nat,F2,A4),aa(B,nat,G,B3)) ) )
         => inj_on(product_prod(A,B),nat,product_case_prod(A,B,nat,aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aga(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F2),G)),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),B4))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_7739_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C3: 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),C3) = 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,C3)),product_Sigma(A,B,J4,C3)) ).

% Sigma_Diff_distrib1
tff(fact_7740_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(A),C3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4),aTP_Lamp_agb(set(B),fun(A,set(B)),C3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B4,aTP_Lamp_agb(set(B),fun(A,set(B)),C3))) ).

% Times_Diff_distrib1
tff(fact_7741_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_agc(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A3)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Diff_distrib2
tff(fact_7742_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_7743_verit__eq__simplify_I20_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X2: 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,X2)) = aa(A,A,aa(num,fun(A,A),F22,X2),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),X2)) ).

% verit_eq_simplify(20)
tff(fact_7744_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ).

% card_cartesian_product
tff(fact_7745_verit__eq__simplify_I21_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X33: 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,X33)) = aa(A,A,aa(num,fun(A,A),F32,X33),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),X33)) ).

% verit_eq_simplify(21)
tff(fact_7746_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C3: set(B),B4: set(A),D4: set(B)] :
      ( 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))),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B4,aTP_Lamp_agb(set(B),fun(A,set(B)),D4))))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( C3 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),D4)) ) ) ) ).

% times_subset_iff
tff(fact_7747_Restr__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B4,aTP_Lamp_agd(set(A),fun(A,set(A)),B4)))),product_Sigma(A,A,A3,aTP_Lamp_agd(set(A),fun(A,set(A)),A3))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agd(set(A),fun(A,set(A)),A3))) ) ) ).

% Restr_subset
tff(fact_7748_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(A),B4: fun(A,set(B)),D4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X3)),aa(A,set(B),D4,X3))) )
       => 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))),product_Sigma(A,B,A3,B4)),product_Sigma(A,B,C3,D4))) ) ) ).

% Sigma_mono
tff(fact_7749_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C3: set(A),A3: set(B),B4: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C3))
     => ( 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))),product_Sigma(B,A,A3,aTP_Lamp_age(set(A),fun(B,set(A)),C3))),product_Sigma(B,A,B4,aTP_Lamp_age(set(A),fun(B,set(A)),C3))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B4)) ) ) ).

% Times_subset_cancel2
tff(fact_7750_open__prod__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( ! [X3: 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)),X3),S))
             => ? [A19: set(A),B15: set(B)] :
                  ( topolo1002775350975398744n_open(A,A19)
                  & topolo1002775350975398744n_open(B,B15)
                  & 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)),X3),product_Sigma(A,B,A19,aTP_Lamp_agf(set(B),fun(A,set(B)),B15))))
                  & 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))),product_Sigma(A,B,A19,aTP_Lamp_agf(set(B),fun(A,set(B)),B15))),S)) ) )
         => topolo1002775350975398744n_open(product_prod(A,B),S) ) ) ).

% open_prod_intro
tff(fact_7751_open__prod__elim,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B)),X: product_prod(A,B)] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S)
         => ( 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)),X),S))
           => ~ ! [A7: set(A)] :
                  ( topolo1002775350975398744n_open(A,A7)
                 => ! [B8: set(B)] :
                      ( topolo1002775350975398744n_open(B,B8)
                     => ( 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)),X),product_Sigma(A,B,A7,aTP_Lamp_agf(set(B),fun(A,set(B)),B8))))
                       => ~ 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))),product_Sigma(A,B,A7,aTP_Lamp_agf(set(B),fun(A,set(B)),B8))),S)) ) ) ) ) ) ) ).

% open_prod_elim
tff(fact_7752_open__prod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S)
        <=> ! [X5: 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)),X5),S))
             => ? [A14: set(A)] :
                  ( topolo1002775350975398744n_open(A,A14)
                  & ? [B10: set(B)] :
                      ( topolo1002775350975398744n_open(B,B10)
                      & 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)),X5),product_Sigma(A,B,A14,aTP_Lamp_agf(set(B),fun(A,set(B)),B10))))
                      & 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))),product_Sigma(A,B,A14,aTP_Lamp_agf(set(B),fun(A,set(B)),B10))),S)) ) ) ) ) ) ).

% open_prod_def
tff(fact_7753_product__atMost__eq__Un,axiom,
    ! [A3: set(nat),M: nat] : product_Sigma(nat,nat,A3,aTP_Lamp_agg(nat,fun(nat,set(nat)),M)) = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),sup_sup(set(product_prod(nat,nat))),product_Sigma(nat,nat,A3,aTP_Lamp_agh(nat,fun(nat,set(nat)),M))),product_Sigma(nat,nat,A3,aTP_Lamp_agi(nat,fun(nat,set(nat)),M))) ).

% product_atMost_eq_Un
tff(fact_7754_subset__fst__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),S: set(product_prod(A,B)),Y4: B] :
      ( 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))),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),B4))),S))
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y4),B4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S))) ) ) ).

% subset_fst_imageI
tff(fact_7755_subset__snd__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),S: set(product_prod(A,B)),X: A] :
      ( 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))),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),B4))),S))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S))) ) ) ).

% subset_snd_imageI
tff(fact_7756_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : 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))),A3),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3),aTP_Lamp_agj(set(product_prod(A,B)),fun(A,set(B)),A3)))) ).

% subset_fst_snd
tff(fact_7757_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] :
    ? [F5: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),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))),aa(set(A),set(product_prod(B,A)),image(A,product_prod(B,A),F5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))),product_Sigma(B,A,I5,A3))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_7758_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hp(nat,fun(nat,fun(nat,bool)),M))) = product_Sigma(nat,nat,set_ord_atMost(nat,M),aTP_Lamp_agh(nat,fun(nat,set(nat)),M)) ).

% pairs_le_eq_Sigma
tff(fact_7759_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F2: fun(A,B),G: fun(C,B),A3: set(A),B4: set(C)] :
          ( inj_on(product_prod(A,C),B,product_case_prod(A,C,B,aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_eq(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F2),G)),product_Sigma(A,C,A3,aTP_Lamp_agk(set(C),fun(A,set(C)),B4)))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B4)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),collect(B,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_agl(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F2),G),A3),B4))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_7760_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B4: set(B)] :
      ( 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))),bNF_Gr(A,B,A3,F2)),product_Sigma(A,B,A3,aTP_Lamp_agb(set(B),fun(A,set(B)),B4))))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4)) ) ).

% Gr_incl
tff(fact_7761_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_ago(fun(nat,rat),fun(nat,rat))),inverse_inverse(real))) ).

% inverse_real.transfer
tff(fact_7762_zero__real_Otransfer,axiom,
    pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,aTP_Lamp_agm(nat,rat)),zero_zero(real))) ).

% zero_real.transfer
tff(fact_7763_one__real_Otransfer,axiom,
    pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,aTP_Lamp_agp(nat,rat)),one_one(real))) ).

% one_real.transfer
tff(fact_7764_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_afv(fun(nat,rat),fun(nat,rat))),uminus_uminus(real))) ).

% uminus_real.transfer
tff(fact_7765_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_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),plus_plus(real))) ).

% plus_real.transfer
tff(fact_7766_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_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),times_times(real))) ).

% times_real.transfer
tff(fact_7767_rat__floor__code,axiom,
    ! [P3: rat] : archim6421214686448440834_floor(rat,P3) = aa(product_prod(int,int),int,product_case_prod(int,int,int,divide_divide(int)),quotient_of(P3)) ).

% rat_floor_code
tff(fact_7768_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_agq(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_7769_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_agr(set(A),fun(nat,fun(list(A),bool)),A3),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_aen(list(A),fun(A,list(A))))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ju(set(A),fun(nat,fun(list(A),bool)),A3),N)),aTP_Lamp_ags(set(A),fun(list(A),set(A)),A3))) ).

% lists_length_Suc_eq
tff(fact_7770_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_agt(fun(nat,rat),bool)),positive2)) ).

% Real.positive.transfer
tff(fact_7771_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_ne(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_ne(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_7772_intrel__iff,axiom,
    ! [X: nat,Y4: nat,U: nat,V: 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),Y4)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,U),V)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y4) ) ) ).

% intrel_iff
tff(fact_7773_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_nf(nat,fun(nat,product_prod(nat,nat))))),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_nf(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_7774_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_7775_int_Oabs__eq__iff,axiom,
    ! [X: product_prod(nat,nat),Y4: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),int,abs_Integ,X) = aa(product_prod(nat,nat),int,abs_Integ,Y4) )
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X),Y4)) ) ).

% int.abs_eq_iff
tff(fact_7776_Real_Opositive__mult,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,positive2,X))
     => ( pp(aa(real,bool,positive2,Y4))
       => pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y4))) ) ) ).

% Real.positive_mult
tff(fact_7777_Real_Opositive__zero,axiom,
    ~ pp(aa(real,bool,positive2,zero_zero(real))) ).

% Real.positive_zero
tff(fact_7778_Real_Opositive__add,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,positive2,X))
     => ( pp(aa(real,bool,positive2,Y4))
       => pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y4))) ) ) ).

% Real.positive_add
tff(fact_7779_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_7780_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_7781_less__real__def,axiom,
    ! [X: real,Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y4))
    <=> pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y4),X))) ) ).

% less_real_def
tff(fact_7782_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_7783_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_ng(nat,fun(nat,A)))),product_case_prod(nat,nat,A,aTP_Lamp_ng(nat,fun(nat,A))))) ) ).

% of_int.rsp
tff(fact_7784_intrel__def,axiom,
    intrel = product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_agv(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).

% intrel_def
tff(fact_7785_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_ni(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ni(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_7786_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_nk(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_nk(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_7787_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_7788_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_nm(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_nm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_7789_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_no(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_no(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_7790_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] :
            ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,X),N4))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_7791_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_agt(fun(nat,rat),bool)),aTP_Lamp_agt(fun(nat,rat),bool))) ).

% Real.positive.rsp
tff(fact_7792_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_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))))) ).

% times_real.rsp
tff(fact_7793_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_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))))) ).

% plus_real.rsp
tff(fact_7794_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_afv(fun(nat,rat),fun(nat,rat))),aTP_Lamp_afv(fun(nat,rat),fun(nat,rat)))) ).

% uminus_real.rsp
tff(fact_7795_zero__real_Orsp,axiom,
    pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,aTP_Lamp_agm(nat,rat)),aTP_Lamp_agm(nat,rat))) ).

% zero_real.rsp
tff(fact_7796_one__real_Orsp,axiom,
    pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,aTP_Lamp_agp(nat,rat)),aTP_Lamp_agp(nat,rat))) ).

% one_real.rsp
tff(fact_7797_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_7798_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_ago(fun(nat,rat),fun(nat,rat))),aTP_Lamp_ago(fun(nat,rat),fun(nat,rat)))) ).

% inverse_real.rsp
tff(fact_7799_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_agt(fun(nat,rat),bool)) ).

% Real.positive_def
tff(fact_7800_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_agm(nat,rat),aTP_Lamp_agn(fun(nat,rat),fun(nat,rat),X))) ) ) ).

% inverse_real.abs_eq
tff(fact_7801_real_Oabs__induct,axiom,
    ! [P: fun(real,bool),X: real] :
      ( ! [Y: fun(nat,rat)] :
          ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,Y),Y))
         => pp(aa(real,bool,P,aa(fun(nat,rat),real,real2,Y))) )
     => pp(aa(real,bool,P,X)) ) ).

% real.abs_induct
tff(fact_7802_of__int__Real,axiom,
    ! [X: int] : aa(int,real,ring_1_of_int(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_agw(int,fun(nat,rat),X)) ).

% of_int_Real
tff(fact_7803_of__rat__Real,axiom,
    ! [X: rat] : aa(rat,real,field_char_0_of_rat(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_afs(rat,fun(nat,rat),X)) ).

% of_rat_Real
tff(fact_7804_one__real__def,axiom,
    one_one(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_agp(nat,rat)) ).

% one_real_def
tff(fact_7805_zero__real__def,axiom,
    zero_zero(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_agm(nat,rat)) ).

% zero_real_def
tff(fact_7806_of__nat__Real,axiom,
    ! [X: nat] : aa(nat,real,semiring_1_of_nat(real),X) = aa(fun(nat,rat),real,real2,aTP_Lamp_agx(nat,fun(nat,rat),X)) ).

% of_nat_Real
tff(fact_7807_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_afv(fun(nat,rat),fun(nat,rat)),X)) ) ) ).

% uminus_real.abs_eq
tff(fact_7808_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_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xa),X)) ) ) ) ).

% plus_real.abs_eq
tff(fact_7809_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_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xa),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_7810_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] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X,N4))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_7811_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_ago(fun(nat,rat),fun(nat,rat))) ).

% inverse_real_def
tff(fact_7812_le__Real,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(fun(nat,rat),real,real2,X7)),aa(fun(nat,rat),real,real2,Y6)))
        <=> ! [R5: rat] :
              ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X7,N4)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y6,N4)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_7813_cr__real__eq,axiom,
    ! [X4: fun(nat,rat),Xa2: real] :
      ( pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),pcr_real,X4),Xa2))
    <=> ( cauchy(X4)
        & ( aa(fun(nat,rat),real,real2,X4) = Xa2 ) ) ) ).

% cr_real_eq
tff(fact_7814_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_7815_cauchy__inverse,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => cauchy(aTP_Lamp_agn(fun(nat,rat),fun(nat,rat),X7)) ) ) ).

% cauchy_inverse
tff(fact_7816_cauchy__add,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6)) ) ) ).

% cauchy_add
tff(fact_7817_cauchy__minus,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_afv(fun(nat,rat),fun(nat,rat)),X7)) ) ).

% cauchy_minus
tff(fact_7818_cauchy__const,axiom,
    ! [X: rat] : cauchy(aTP_Lamp_afs(rat,fun(nat,rat),X)) ).

% cauchy_const
tff(fact_7819_cauchy__mult,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6)) ) ) ).

% cauchy_mult
tff(fact_7820_cauchy__diff,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6)) ) ) ).

% cauchy_diff
tff(fact_7821_realrel__refl,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X7),X7)) ) ).

% realrel_refl
tff(fact_7822_cauchy__imp__bounded,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ? [B3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B3))
          & ! [N7: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X7,N7))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_7823_less__RealD,axiom,
    ! [Y6: fun(nat,rat),X: real] :
      ( cauchy(Y6)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(fun(nat,rat),real,real2,Y6)))
       => ? [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,Y6,N2)))) ) ) ).

% less_RealD
tff(fact_7824_Real__leI,axiom,
    ! [X7: fun(nat,rat),Y4: real] :
      ( cauchy(X7)
     => ( ! [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,X7,N2))),Y4))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(fun(nat,rat),real,real2,X7)),Y4)) ) ) ).

% Real_leI
tff(fact_7825_le__RealI,axiom,
    ! [Y6: fun(nat,rat),X: real] :
      ( cauchy(Y6)
     => ( ! [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,Y6,N2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(fun(nat,rat),real,real2,Y6))) ) ) ).

% le_RealI
tff(fact_7826_minus__Real,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,X7)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_afv(fun(nat,rat),fun(nat,rat)),X7)) ) ) ).

% minus_Real
tff(fact_7827_add__Real,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,X7)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6)) ) ) ) ).

% add_Real
tff(fact_7828_mult__Real,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,X7)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X7),Y6)) ) ) ) ).

% mult_Real
tff(fact_7829_diff__Real,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(fun(nat,rat),real,real2,X7)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6)) ) ) ) ).

% diff_Real
tff(fact_7830_realrelI,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6)))
         => pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X7),Y6)) ) ) ) ).

% realrelI
tff(fact_7831_eq__Real,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( cauchy(Y6)
       => ( ( aa(fun(nat,rat),real,real2,X7) = aa(fun(nat,rat),real,real2,Y6) )
        <=> pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6))) ) ) ) ).

% eq_Real
tff(fact_7832_vanishes__diff__inverse,axiom,
    ! [X7: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ( cauchy(Y6)
         => ( ~ pp(vanishes(Y6))
           => ( pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6)))
             => pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_agy(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X7),Y6))) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_7833_realrel__def,axiom,
    ! [X4: fun(nat,rat),Xa2: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X4),Xa2))
    <=> ( cauchy(X4)
        & cauchy(Xa2)
        & pp(vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X4),Xa2))) ) ) ).

% realrel_def
tff(fact_7834_cauchy__not__vanishes__cases,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B3))
            & ? [K2: nat] :
                ( ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N7))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B3),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X7,N7)))) )
                | ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N7))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B3),aa(nat,rat,X7,N7))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_7835_positive__Real,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( pp(aa(real,bool,positive2,aa(fun(nat,rat),real,real2,X7)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X7,N4))) ) ) ) ) ).

% positive_Real
tff(fact_7836_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_afv(fun(nat,rat),fun(nat,rat))) ).

% uminus_real_def
tff(fact_7837_cauchy__not__vanishes,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ~ pp(vanishes(X7))
       => ? [B3: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B3))
            & ? [K2: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N7))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B3),aa(rat,rat,abs_abs(rat),aa(nat,rat,X7,N7)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_7838_cauchyD,axiom,
    ! [X7: fun(nat,rat),R2: rat] :
      ( cauchy(X7)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
           => ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N7))
               => 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,X7,M2)),aa(nat,rat,X7,N7)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_7839_cauchyI,axiom,
    ! [X7: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K4: nat] :
            ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),M4))
             => ! [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,X7,M4)),aa(nat,rat,X7,N2)))),R3)) ) ) )
     => cauchy(X7) ) ).

% cauchyI
tff(fact_7840_cauchy__def,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
    <=> ! [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))
             => ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),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,X7,M5)),aa(nat,rat,X7,N4)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_7841_inverse__Real,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ( pp(vanishes(X7))
         => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X7)) = zero_zero(real) ) )
        & ( ~ pp(vanishes(X7))
         => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X7)) = aa(fun(nat,rat),real,real2,aTP_Lamp_agn(fun(nat,rat),fun(nat,rat),X7)) ) ) ) ) ).

% inverse_Real
tff(fact_7842_not__positive__Real,axiom,
    ! [X7: fun(nat,rat)] :
      ( cauchy(X7)
     => ( ~ pp(aa(real,bool,positive2,aa(fun(nat,rat),real,real2,X7)))
      <=> ! [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
           => ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X7,N4)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_7843_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_afr(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real_def
tff(fact_7844_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_afu(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real_def
tff(fact_7845_cr__real__def,axiom,
    ! [X4: fun(nat,rat),Xa2: real] :
      ( pp(aa(real,bool,aa(fun(nat,rat),fun(real,bool),cr_real,X4),Xa2))
    <=> ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X4),X4))
        & ( aa(fun(nat,rat),real,real2,X4) = Xa2 ) ) ) ).

% cr_real_def
tff(fact_7846_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A3: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] :
          ( bi_total(A,B,A3)
         => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A3,bNF_rel_fun(A,B,bool,bool,A3,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
           => ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B4,bNF_rel_fun(C,D,bool,bool,B4,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
             => pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A3,B4),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).

% mono_transfer
tff(fact_7847_int_Obi__total,axiom,
    bi_total(product_prod(nat,nat),int,pcr_int) ).

% int.bi_total
tff(fact_7848_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_7849_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( 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),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aha(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_7850_rcis__inverse,axiom,
    ! [R2: real,A2: real] : aa(complex,complex,inverse_inverse(complex),rcis(R2,A2)) = rcis(aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),R2),aa(real,real,uminus_uminus(real),A2)) ).

% rcis_inverse
tff(fact_7851_rcis__zero__arg,axiom,
    ! [R2: real] : rcis(R2,zero_zero(real)) = real_Vector_of_real(complex,R2) ).

% rcis_zero_arg
tff(fact_7852_rcis__eq__zero__iff,axiom,
    ! [R2: real,A2: real] :
      ( ( rcis(R2,A2) = zero_zero(complex) )
    <=> ( R2 = zero_zero(real) ) ) ).

% rcis_eq_zero_iff
tff(fact_7853_rcis__zero__mod,axiom,
    ! [A2: real] : rcis(zero_zero(real),A2) = zero_zero(complex) ).

% rcis_zero_mod
tff(fact_7854_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( 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),R))) ) ) ).

% finite_relpow
tff(fact_7855_Re__rcis,axiom,
    ! [R2: real,A2: real] : re(rcis(R2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),cos(real,A2)) ).

% Re_rcis
tff(fact_7856_Im__rcis,axiom,
    ! [R2: real,A2: real] : im(rcis(R2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),sin(real,A2)) ).

% Im_rcis
tff(fact_7857_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y4: A,R: 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),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R)))
     => ( X = Y4 ) ) ).

% relpow_0_E
tff(fact_7858_relpow__0__I,axiom,
    ! [A: $tType,X: A,R: 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)),R))) ).

% relpow_0_I
tff(fact_7859_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y4: A,R: 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),Y4)),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,Y4),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),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),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)),R))) ) ) ).

% relpow_Suc_I2
tff(fact_7860_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: 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)),R)))
     => ~ ! [Y: 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),Y)),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,Y),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R))) ) ) ).

% relpow_Suc_E2
tff(fact_7861_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: 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)),R)))
     => ? [Y: 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),Y)),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,Y),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R))) ) ) ).

% relpow_Suc_D2
tff(fact_7862_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y4: A,N: nat,R: 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),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),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,Y4),Z)),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),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)),R))) ) ) ).

% relpow_Suc_I
tff(fact_7863_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: 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)),R)))
     => ~ ! [Y: 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),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),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,Y),Z)),R)) ) ) ).

% relpow_Suc_E
tff(fact_7864_relpow__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: 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),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( 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),Y)),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,Y),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M4),R))) ) ) ) ) ).

% relpow_E2
tff(fact_7865_relpow__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: 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),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( 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),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M4),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,Y),Z)),R)) ) ) ) ) ).

% relpow_E
tff(fact_7866_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_7867_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),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))),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),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_ahb(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ).

% relpow_finite_bounded
tff(fact_7868_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,N: nat,R: 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,A2),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F7,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,F7,I2)),aa(nat,A,F7,aa(nat,nat,suc,I2)))),R)) ) ) ) ).

% relpow_fun_conv
tff(fact_7869_rcis__mult,axiom,
    ! [R12: real,A2: real,R23: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),rcis(R12,A2)),rcis(R23,B2)) = rcis(aa(real,real,aa(real,fun(real,real),times_times(real),R12),R23),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)) ).

% rcis_mult
tff(fact_7870_rcis__divide,axiom,
    ! [R12: real,A2: real,R23: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),rcis(R12,A2)),rcis(R23,B2)) = rcis(aa(real,real,aa(real,fun(real,real),divide_divide(real),R12),R23),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)) ).

% rcis_divide
tff(fact_7871_DeMoivre2,axiom,
    ! [R2: real,A2: real,N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),rcis(R2,A2)),N) = rcis(aa(nat,real,aa(real,fun(nat,real),power_power(real),R2),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) ).

% DeMoivre2
tff(fact_7872_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,N,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_ahc(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_7873_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( transitive_trancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aha(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_7874_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R) = R ).

% ntrancl_Zero
tff(fact_7875_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( transitive_trancl(A,R) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R)),one_one(nat)),R) ) ) ).

% finite_trancl_ntranl
tff(fact_7876_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R: 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)),P3),transitive_trancl(A,R)))
    <=> ? [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
          & 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)),P3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N4),R))) ) ) ).

% trancl_power
tff(fact_7877_trancl__subset__Sigma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( 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))),R2),product_Sigma(A,A,A3,aTP_Lamp_agd(set(A),fun(A,set(A)),A3))))
     => 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))),transitive_trancl(A,R2)),product_Sigma(A,A,A3,aTP_Lamp_agd(set(A),fun(A,set(A)),A3)))) ) ).

% trancl_subset_Sigma
tff(fact_7878_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A)),S2: 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)),P3),transitive_trancl(A,R2)))
     => ( 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))),R2),S2))
       => 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)),P3),transitive_trancl(A,S2))) ) ) ).

% trancl_mono
tff(fact_7879_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_7880_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_ahb(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_7881_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),A2),B2))
    <=> ? [Z5: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,aa(list(product_prod(A,B)),fun(set(list(product_prod(A,B))),bool),member(list(product_prod(A,B))),Z5),collect(list(product_prod(A,B)),aTP_Lamp_ahd(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R))))
          & ( map(product_prod(A,B),A,product_fst(A,B),Z5) = A2 )
          & ( map(product_prod(A,B),B,product_snd(A,B),Z5) = B2 ) ) ) ).

% list.in_rel
tff(fact_7882_rtrancl__Un__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] : 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(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),transitive_rtrancl(A,S))),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S)))) ).

% rtrancl_Un_subset
tff(fact_7883_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),Ra))
     => pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),list_all2(A,B,R)),list_all2(A,B,Ra))) ) ).

% list.rel_mono
tff(fact_7884_rtrancl__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( 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))),R2),S2))
     => 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))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_mono
tff(fact_7885_rtrancl__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( 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),S))
     => ( 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))),S),transitive_rtrancl(A,R)))
       => ( transitive_rtrancl(A,S) = transitive_rtrancl(A,R) ) ) ) ).

% rtrancl_subset
tff(fact_7886_rtrancl__subset__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( 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))),R2),transitive_rtrancl(A,S2)))
     => 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))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_subset_rtrancl
tff(fact_7887_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),A3: set(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,A2),B2)),transitive_rtrancl(A,R2)))
     => ( 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))),R2),product_Sigma(A,A,A3,aTP_Lamp_agd(set(A),fun(A,set(A)),A3))))
       => ( ( A2 = B2 )
          | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7888_pred__nat__trancl__eq__le,axiom,
    ! [M: 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,M),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_7889_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A3: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A3,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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),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,A3),A3),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).

% sum_list_transfer
tff(fact_7890_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A3: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A3,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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),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,B4,A3),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A3,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B4),A3))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_7891_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A3: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A3,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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),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,A3),A3),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).

% prod_list_transfer
tff(fact_7892_power_Opower__eq__if,axiom,
    ! [A: $tType,M: nat,One: A,Times: fun(A,fun(A,A)),P3: A] :
      ( ( ( M = zero_zero(nat) )
       => ( power2(A,One,Times,P3,M) = One ) )
      & ( ( M != zero_zero(nat) )
       => ( power2(A,One,Times,P3,M) = aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_7893_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),cons(A,X,Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).

% prod_list.Cons
tff(fact_7894_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),append(A,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ).

% prod_list.append
tff(fact_7895_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_7896_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A,N: nat] : power2(A,One,Times,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),Times,A2),power2(A,One,Times,A2,N)) ).

% power.power.power_Suc
tff(fact_7897_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A] : power2(A,One,Times,A2,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_7898_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_7899_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_7900_remove__def,axiom,
    ! [A: $tType,X: A,A3: set(A)] : remove(A,X,A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% remove_def
tff(fact_7901_member__remove,axiom,
    ! [A: $tType,X: A,Y4: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),remove(A,Y4,A3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & ( X != Y4 ) ) ) ).

% member_remove
tff(fact_7902_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_7903_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_7904_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_7905_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_7906_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_7907_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
        <=> ! [S9: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S9),A3))
             => ( pp(aa(set(A),bool,finite_finite(A),S9))
               => ! [U6: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),U6)),S9) = zero_zero(A) )
                   => ! [X5: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S9))
                       => ( aa(A,real,U6,X5) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_7908_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( ~ real_V358717886546972837endent(A,S2)
        <=> ! [T3: set(A),U6: fun(A,real),V6: A] :
              ( pp(aa(set(A),bool,finite_finite(A),T3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),U6)),T3) = zero_zero(A) )
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),T3))
                   => ( aa(A,real,U6,V6) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_7909_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_7910_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A3))
         => real_V358717886546972837endent(A,A3) ) ) ).

% dependent_zero
tff(fact_7911_independent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
           => ~ real_V358717886546972837endent(A,B4) ) ) ) ).

% independent_mono
tff(fact_7912_dependent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),A3: set(A)] :
          ( real_V358717886546972837endent(A,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
           => real_V358717886546972837endent(A,A3) ) ) ) ).

% dependent_mono
tff(fact_7913_independent__Union__directed,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C3: set(set(A))] :
          ( ! [C4: set(A),D3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
             => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),D3),C3))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),D3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D3),C4)) ) ) )
         => ( ! [C4: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
               => ~ real_V358717886546972837endent(A,C4) )
           => ~ real_V358717886546972837endent(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ).

% independent_Union_directed
tff(fact_7914_unique__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( ! [V4: A] :
                ( ( aa(A,real,F2,V4) != zero_zero(real) )
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V4),Basis)) )
           => ( ! [V4: A] :
                  ( ( aa(A,real,G,V4) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V4),Basis)) )
             => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F2))))
               => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),G))))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),F2)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F2))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),G)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),G))) )
                   => ( F2 = G ) ) ) ) ) ) ) ) ).

% unique_representation
tff(fact_7915_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [F5: fun(A,real),X3: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),F5)),A3) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => ( aa(A,real,F5,X3) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A3) ) ) ) ).

% independent_if_scalars_zero
tff(fact_7916_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( real_V358717886546972837endent(A,S)
          <=> ? [U6: fun(A,real)] :
                ( ? [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S))
                    & ( aa(A,real,U6,X5) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),U6)),S) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_7917_independentD__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),X7: fun(A,real),Y6: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,B4)
         => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(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_ahf(fun(A,real),fun(A,bool),X7))),B4))
             => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Y6))))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Y6))),B4))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X7))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),Y6)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Y6))) )
                   => ( X7 = Y6 ) ) ) ) ) ) ) ) ).

% independentD_unique
tff(fact_7918_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A),T2: set(A),U: fun(A,real),V: A] :
          ( ~ real_V358717886546972837endent(A,S2)
         => ( 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),S2))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),U)),T2) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),T2))
                 => ( aa(A,real,U,V) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_7919_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A)] :
          ( real_V358717886546972837endent(A,B4)
        <=> ? [X9: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))),B4))
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),X9)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))) = zero_zero(A) )
              & ? [X5: A] : aa(A,real,X9,X5) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_7920_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A)] :
          ( ~ real_V358717886546972837endent(A,B4)
        <=> ! [X9: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))),B4))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),X9)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X9))) = zero_zero(A) )
                 => ! [X5: A] : aa(A,real,X9,X5) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_7921_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),X7: fun(A,real),X: A] :
          ( ~ real_V358717886546972837endent(A,B4)
         => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(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_ahf(fun(A,real),fun(A,bool),X7))),B4))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),X7))) = zero_zero(A) )
               => ( aa(A,real,X7,X) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_7922_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( real_V358717886546972837endent(A,S2)
        <=> ? [T3: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),T3))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
              & ? [U6: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),U6)),T3) = zero_zero(A) )
                  & ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),T3))
                      & ( aa(A,real,U6,X5) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_7923_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X7: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X7)))
             => ( infini527867602293511546merate(A,X7,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite(A),X7))
           => ( pp(aa(set(A),bool,finite_finite(A),Y6))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X7)),aa(set(A),nat,finite_card(A),Y6)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),Y6)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_7924_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X5: A,Y5: A] :
                      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X5,Y5)),S6))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X5),aa(A,B,F2,Y5))),R5)) ) ) ) ) ) ).

% isUCont_def
tff(fact_7925_le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N))) ) ).

% le_enumerate
tff(fact_7926_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),S))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N)))) ) ) ).

% enumerate_step
tff(fact_7927_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S2: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S2,F2)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [D5: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
                  & ! [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S2))
                     => ! [Xa4: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Xa4,X5)),D5))
                           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa4),aa(A,B,F2,X5))),E3)) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_7928_finite__le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N))) ) ) ).

% finite_le_enumerate
tff(fact_7929_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N)))) ) ) ) ).

% finite_enumerate_step
tff(fact_7930_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(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,infini527867602293511546merate(A,S,zero_zero(nat))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc'
tff(fact_7931_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S)))
           => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ahg(set(A),fun(nat,fun(A,bool)),S),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_7932_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(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,ord_Least(A,aTP_Lamp_ahh(set(A),fun(A,bool),S))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc
tff(fact_7933_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_7934_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI
tff(fact_7935_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A2))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2
tff(fact_7936_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI_ex
tff(fact_7937_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_ex
tff(fact_7938_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_7939_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,Q,M))
       => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
         => ( ! [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_7940_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_aaa(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_7941_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),K)) ) ) ).

% Least_le
tff(fact_7942_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B9: A] :
                      ( pp(aa(A,bool,P,B9))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B9)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7943_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A2))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B9: A] :
                      ( pp(aa(A,bool,P,B9))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B9)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder
tff(fact_7944_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y: A] :
                ( pp(aa(A,bool,P,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_7945_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y: A] :
                ( pp(aa(A,bool,P,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).

% LeastI2_order
tff(fact_7946_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),Z: A] :
          ( ? [X4: A] :
              ( pp(aa(A,bool,P,X4))
              & ! [Y: A] :
                  ( pp(aa(A,bool,P,Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y)) )
              & ! [Y: A] :
                  ( ( pp(aa(A,bool,P,Y))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Ya2)) ) )
                 => ( Y = X4 ) ) )
         => ( pp(aa(A,bool,P,Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),Z)) ) ) ) ).

% Least1_le
tff(fact_7947_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] :
          ( ? [X4: A] :
              ( pp(aa(A,bool,P,X4))
              & ! [Y: A] :
                  ( pp(aa(A,bool,P,Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y)) )
              & ! [Y: A] :
                  ( ( pp(aa(A,bool,P,Y))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Ya2)) ) )
                 => ( Y = X4 ) ) )
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% Least1I
tff(fact_7948_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] : infini527867602293511546merate(A,S,zero_zero(nat)) = ord_Least(A,aTP_Lamp_ahh(set(A),fun(A,bool),S)) ) ).

% enumerate_0
tff(fact_7949_Sup__real__def,axiom,
    ! [X7: set(real)] : aa(set(real),real,complete_Sup_Sup(real),X7) = ord_Least(real,aTP_Lamp_ahi(set(real),fun(real,bool),X7)) ).

% Sup_real_def
tff(fact_7950_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                & ! [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa3)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_ahj(fun(A,B),fun(set(A),fun(B,bool)),F2),S)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_ahk(set(A),fun(A,bool),S))) ) ) ) ) ).

% Least_mono
tff(fact_7951_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_ahl(fun(A,bool),fun(A,bool),P)) ) ).

% Least_def
tff(fact_7952_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),S))
         => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ahg(set(A),fun(nat,fun(A,bool)),S),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_7953_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_7954_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A3) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_7955_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J)))
        <=> ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
            | pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).

% possible_bit_min
tff(fact_7956_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_7957_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_gcd_Gcd_fin(A,A3)),one_one(A)))
        <=> ( semiring_gcd_Gcd_fin(A,A3) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_7958_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B4: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,B4)),semiring_gcd_Gcd_fin(A,A3)) = semiring_gcd_Gcd_fin(A,A3) ) ) ) ).

% Gcd_fin.subset
tff(fact_7959_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
           => pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).

% possible_bit_less_imp
tff(fact_7960_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_7961_Gcd__fin__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_gcd_Gcd_fin(A,A3)),A2)) ) ) ).

% Gcd_fin_dvd
tff(fact_7962_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),semiring_gcd_Gcd_fin(A,A3))) ) ) ) ).

% Gcd_fin_greatest
tff(fact_7963_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),semiring_gcd_Gcd_fin(A,A3)))
          <=> ! [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),X5)) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_7964_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A2: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),aa(list(A),set(A),set2(A),Bs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Bs)))) ) ) ).

% gcd_list_greatest
tff(fact_7965_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,Xs: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xs))))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),X5)) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_7966_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),insert(A,A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7967_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => ( semiring_gcd_Gcd_fin(A,A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7968_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,M),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),M),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% drop_bit_exp_eq
tff(fact_7969_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_7970_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7971_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_7972_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_7973_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff
tff(fact_7974_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% bit_imp_possible_bit
tff(fact_7975_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N))
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% impossible_bit
tff(fact_7976_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ! [N4: nat] :
              ( pp(bit_se6407376104438227557le_bit(A,type2(A),N4))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N4))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N4)) ) ) ) ) ).

% bit_eq_iff
tff(fact_7977_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( pp(bit_se6407376104438227557le_bit(A,type2(A),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) ) )
         => ( A2 = B2 ) ) ) ).

% bit_eqI
tff(fact_7978_bit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            | ( ( M = N )
              & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ) ).

% bit_set_bit_iff
tff(fact_7979_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)),N))
        <=> ( ( ( M = N )
            <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_flip_bit_iff
tff(fact_7980_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_iff
tff(fact_7981_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_7982_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X3: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X3) = zero_zero(A) )
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C2),X3)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7983_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(int,A,ring_1_of_int(A),K)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ) ) ).

% bit_of_int_iff
tff(fact_7984_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N)) ) ) ) ).

% bit_of_nat_iff
tff(fact_7985_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_7986_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),N)) ) ) ) ).

% bit_minus_iff
tff(fact_7987_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))))
      <=> ? [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
            & ( aa(nat,A,semiring_1_of_nat(A),N4) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7988_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) )
           => ( ! [X3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X3))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),C2))
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7989_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
           => ( aa(nat,A,semiring_1_of_nat(A),N4) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7990_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,M),A2)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_push_bit_iff
tff(fact_7991_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_7992_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ( M = N ) ) ) ) ).

% bit_exp_iff
tff(fact_7993_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_7994_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ( N != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_7995_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7996_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,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))),M)),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),M)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7997_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),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_7998_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_7999_connected__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1966860045006549960nected(A,S2)
        <=> ~ ? [A14: set(A),B10: set(A)] :
                ( topolo7761053866217962861closed(A,A14)
                & topolo7761053866217962861closed(A,B10)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A14),B10)))
                & ( 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)),A14),B10)),S2) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A14),S2) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B10),S2) != bot_bot(set(A)) ) ) ) ) ).

% connected_closed
tff(fact_8000_last__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,replicate(A,N,X)) = X ) ) ).

% last_replicate
tff(fact_8001_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_8002_connected__contains__Ioo,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),A3)) ) ) ) ) ).

% connected_contains_Ioo
tff(fact_8003_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U3: set(A)] :
          ( topolo1966860045006549960nected(A,U3)
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),U3))
             => ! [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),U3))
                 => ! [Z5: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Z5))
                     => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z5),Xa4))
                       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z5),U3)) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_8004_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U3: set(A)] :
          ( ! [X3: A,Y: A,Z2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),U3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),U3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),U3)) ) ) ) )
         => topolo1966860045006549960nected(A,U3) ) ) ).

% connectedI_interval
tff(fact_8005_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U3: set(A),X: A,Y4: A,Z: A] :
          ( topolo1966860045006549960nected(A,U3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),U3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),U3))
             => ( 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),Z),Y4))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),U3)) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_8006_connected__contains__Icc,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),A3)) ) ) ) ) ).

% connected_contains_Icc
tff(fact_8007_connected__diff__open__from__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T2: set(A),U: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),U))
           => ( topolo1002775350975398744n_open(A,S2)
             => ( topolo7761053866217962861closed(A,T2)
               => ( topolo1966860045006549960nected(A,U)
                 => ( topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                   => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),U),S2)) ) ) ) ) ) ) ) ).

% connected_diff_open_from_closed
tff(fact_8008_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_8009_connectedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(A),U3: set(A),V2: set(A)] :
          ( topolo1966860045006549960nected(A,A3)
         => ( topolo1002775350975398744n_open(A,U3)
           => ( topolo1002775350975398744n_open(A,V2)
             => ( ( 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)),U3),V2)),A3) = bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),U3),V2)))
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U3),A3) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),V2),A3) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connectedD
tff(fact_8010_connectedI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U3: set(A)] :
          ( ! [A7: set(A)] :
              ( topolo1002775350975398744n_open(A,A7)
             => ! [B8: set(A)] :
                  ( topolo1002775350975398744n_open(A,B8)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),U3) != bot_bot(set(A)) )
                   => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B8),U3) != bot_bot(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)),A7),B8)),U3) = bot_bot(set(A)) )
                       => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),U3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B8))) ) ) ) ) )
         => topolo1966860045006549960nected(A,U3) ) ) ).

% connectedI
tff(fact_8011_connected__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1966860045006549960nected(A,S)
        <=> ~ ? [A14: set(A),B10: set(A)] :
                ( topolo1002775350975398744n_open(A,A14)
                & topolo1002775350975398744n_open(A,B10)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A14),B10)))
                & ( 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)),A14),B10)),S) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A14),S) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B10),S) != bot_bot(set(A)) ) ) ) ) ).

% connected_def
tff(fact_8012_connected__closedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),A3: set(A),B4: set(A)] :
          ( topolo1966860045006549960nected(A,S2)
         => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),S2) = bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)))
             => ( topolo7761053866217962861closed(A,A3)
               => ( topolo7761053866217962861closed(A,B4)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S2) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),S2) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connected_closedD
tff(fact_8013_minus__coset__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),coset(A,Xs)) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_a(set(A),fun(A,bool),A3),Xs)) ).

% minus_coset_filter
tff(fact_8014_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] : 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_ahm(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),vEBT_rec_VEBT(A,F1,F22,X14)) ).

% VEBT.simps(7)
tff(fact_8015_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,X222: bool] : vEBT_rec_VEBT(A,F1,F22,vEBT_Leaf(X21,X222)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X222) ).

% VEBT.simps(8)
tff(fact_8016_subset__code_I2_J,axiom,
    ! [B: $tType,A3: set(B),Ys: list(B)] :
      ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),coset(B,Ys)))
    <=> ! [X5: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),aa(list(B),set(B),set2(B),Ys)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A3)) ) ) ).

% subset_code(2)
tff(fact_8017_subset__code_I3_J,axiom,
    ! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).

% subset_code(3)
tff(fact_8018_flat__lub__def,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A3) = B2 ) )
      & ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A3) = the(A,aa(A,fun(A,bool),aTP_Lamp_ahn(set(A),fun(A,fun(A,bool)),A3),B2)) ) ) ) ).

% flat_lub_def
tff(fact_8019_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( inj_on(A,B,F2,collect(A,P))
         => ( pp(aa(A,bool,P,A2))
           => ( ! [Y: A] :
                  ( pp(aa(A,bool,P,Y))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),aa(A,B,F2,Y))) )
             => ( lattices_ord_arg_min(A,B,F2,P) = A2 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_8020_arg__min__nat__le,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,X))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,X))) ) ).

% arg_min_nat_le
tff(fact_8021_arg__min__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ( pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P)))
        & ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Y3))) ) ) ) ).

% arg_min_nat_lemma
tff(fact_8022_arg__min__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P: fun(C,bool),K: C,F2: fun(C,A)] :
          ( pp(aa(C,bool,P,K))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F2,K)),aa(C,A,F2,X3))) )
           => ( aa(C,A,F2,lattices_ord_arg_min(C,A,F2,P)) = aa(C,A,F2,K) ) ) ) ) ).

% arg_min_equality
tff(fact_8023_chains__extend,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A)),Z: set(A)] :
      ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),C2),chains(A,S)))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Z),S))
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C2))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Z)) )
         => pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),insert(set(A),Z),bot_bot(set(set(A))))),C2)),chains(A,S))) ) ) ) ).

% chains_extend
tff(fact_8024_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,X),A3)) = 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_8025_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A16),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_8026_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_8027_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X))
            <=> ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_8028_chainsD2,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A))] :
      ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),C2),chains(A,S)))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C2),S)) ) ).

% chainsD2
tff(fact_8029_chainsD,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A)),X: set(A),Y4: set(A)] :
      ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),C2),chains(A,S)))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X),C2))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y4),C2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X),Y4))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y4),X)) ) ) ) ) ).

% chainsD
tff(fact_8030_Zorn__Lemma2,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),X3),chains(A,A3)))
         => ? [Xa2: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa2),A3))
              & ! [Xb2: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xb2),X3))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xb2),Xa2)) ) ) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
          & ! [Xa2: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa2),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa2))
               => ( Xa2 = X3 ) ) ) ) ) ).

% Zorn_Lemma2
tff(fact_8031_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),lattic5882676163264333800up_fin(A,A3))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_8032_Zorn__Lemma,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),X3),chains(A,A3)))
         => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X3)),A3)) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
          & ! [Xa2: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa2),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa2))
               => ( Xa2 = X3 ) ) ) ) ) ).

% Zorn_Lemma
tff(fact_8033_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B4))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_8034_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,B4)),lattic5882676163264333800up_fin(A,A3)) = lattic5882676163264333800up_fin(A,A3) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_8035_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic5882676163264333800up_fin(A,A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic5882676163264333800up_fin(A,A3) = 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_8036_chains__def,axiom,
    ! [A: $tType,A3: set(set(A))] : chains(A,A3) = collect(set(set(A)),aTP_Lamp_aho(set(set(A)),fun(set(set(A)),bool),A3)) ).

% chains_def
tff(fact_8037_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool,aTP_Lamp_ahp(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_8038_chain__subset__def,axiom,
    ! [A: $tType,C3: set(set(A))] :
      ( chain_subset(A,C3)
    <=> ! [X5: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),C3))
         => ! [Xa4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa4),C3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),Xa4))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa4),X5)) ) ) ) ) ).

% chain_subset_def
tff(fact_8039_initial__segment__of__Diff,axiom,
    ! [A: $tType,P3: set(product_prod(A,A)),Q2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A)),P3),Q2)),init_seg_of(A)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P3),S2)),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))),Q2),S2))),init_seg_of(A))) ) ).

% initial_segment_of_Diff
tff(fact_8040_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,X),A3)) = 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_8041_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( lattic7752659483105999362nf_fin(A,A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( lattic7752659483105999362nf_fin(A,A3) = 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)),A3),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_8042_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3)))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A16)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_8043_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_8044_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3)))
            <=> ! [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_8045_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A3)),A2)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_8046_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B4)),lattic7752659483105999362nf_fin(A,A3))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_8047_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,B4)),lattic7752659483105999362nf_fin(A,A3)) = lattic7752659483105999362nf_fin(A,A3) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_8048_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A3)),lattic5882676163264333800up_fin(A,A3))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_8049_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A2: B] :
      ( inj_on(A,B,F2,A3)
     => 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)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))),A3)),aa(set(A),set(A),insert(A,the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahq(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F2),A3),A2))),bot_bot(set(A))))) ) ).

% inj_on_vimage_singleton
tff(fact_8050_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))),aa(set(A),set(A),insert(A,the(A,aa(B,fun(A,bool),aTP_Lamp_ahr(fun(A,B),fun(B,fun(A,bool)),F2),A2))),bot_bot(set(A))))) ) ).

% inj_vimage_singleton
tff(fact_8051_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,B4)))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),B4)) ) ).

% vimage_eq
tff(fact_8052_vimageI,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,B2: A,B4: set(A)] :
      ( ( aa(B,A,F2,A2) = B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B4))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),vimage(B,A,F2,B4))) ) ) ).

% vimageI
tff(fact_8053_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),P: fun(B,bool)] : vimage(A,B,F2,collect(B,P)) = collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_ahs(fun(A,B),fun(fun(B,bool),fun(A,bool)),F2),P)) ).

% vimage_Collect_eq
tff(fact_8054_vimage__ident,axiom,
    ! [A: $tType,Y6: set(A)] : vimage(A,A,aTP_Lamp_nt(A,A),Y6) = Y6 ).

% vimage_ident
tff(fact_8055_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,top_top(set(B))) = top_top(set(A)) ).

% vimage_UNIV
tff(fact_8056_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_8057_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B4)) ).

% vimage_Int
tff(fact_8058_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B4)) ).

% vimage_Un
tff(fact_8059_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,A3: set(B)] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( vimage(A,B,aTP_Lamp_nz(B,fun(A,B),C2),A3) = top_top(set(A)) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( vimage(A,B,aTP_Lamp_nz(B,fun(A,B),C2),A3) = bot_bot(set(A)) ) ) ) ).

% vimage_const
tff(fact_8060_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(A)] : aa(set(B),set(A),image(B,A,F2),vimage(B,A,F2,A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_8061_vimage__if,axiom,
    ! [B: $tType,A: $tType,C2: B,A3: set(B),D2: B,B4: set(A)] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aht(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B4),A3) = top_top(set(A)) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aht(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B4),A3) = B4 ) ) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aht(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B4),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B4) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aht(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B4),A3) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_8062_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),vimage(B,A,F2,A3))),A3)) ).

% image_vimage_subset
tff(fact_8063_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B4))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),vimage(B,A,F2,B4))) ) ).

% image_subset_iff_subset_vimage
tff(fact_8064_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),B4: set(A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),vimage(B,A,F2,B4)),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),A3))) ) ) ).

% vimage_subsetD
tff(fact_8065_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),vimage(A,B,F2,B4)))
    <=> ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X5)),B4)) ) ) ).

% subset_vimage_iff
tff(fact_8066_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),vimage(B,A,F2,A3)),vimage(B,A,F2,B4))) ) ).

% vimage_mono
tff(fact_8067_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: B,B4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),insert(B,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))),vimage(A,B,F2,B4)) ).

% vimage_insert
tff(fact_8068_continuous__imp__open__vimage,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F2: fun(A,B),B4: set(B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( topolo1002775350975398744n_open(B,B4)
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B4)),S2))
               => topolo1002775350975398744n_open(A,vimage(A,B,F2,B4)) ) ) ) ) ) ).

% continuous_imp_open_vimage
tff(fact_8069_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,B),G: fun(A,B),Y4: set(B)] :
      ( ! [W2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W2),S))
         => ( aa(A,B,F2,W2) = aa(A,B,G,W2) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Y4)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Y4)),S) ) ) ).

% vimage_inter_cong
tff(fact_8070_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B4)) ).

% vimage_Diff
tff(fact_8071_vimage__Suc__insert__0,axiom,
    ! [A3: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,zero_zero(nat)),A3)) = vimage(nat,nat,suc,A3) ).

% vimage_Suc_insert_0
tff(fact_8072_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F2: fun(A,B),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(B,bool,P,aa(A,B,F2,X3)))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( vimage(A,B,F2,collect(B,P)) = collect(A,Q) ) ) ).

% vimage_Collect
tff(fact_8073_vimageI2,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,A2)),A3))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),vimage(B,A,F2,A3))) ) ).

% vimageI2
tff(fact_8074_vimageE,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,B4)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),B4)) ) ).

% vimageE
tff(fact_8075_vimageD,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,A3)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),A3)) ) ).

% vimageD
tff(fact_8076_vimage__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B4: set(B)] : vimage(A,B,F2,B4) = collect(A,aa(set(B),fun(A,bool),aTP_Lamp_ahu(fun(A,B),fun(set(B),fun(A,bool)),F2),B4)) ).

% vimage_def
tff(fact_8077_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A3)) ).

% vimage_Compl
tff(fact_8078_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A3: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),A3)) = aa(set(nat),set(nat),insert(nat,N),vimage(nat,nat,suc,A3)) ).

% vimage_Suc_insert_Suc
tff(fact_8079_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))))
    <=> ( aa(A,B,F2,A2) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_8080_finite__vimage__Suc__iff,axiom,
    ! [F3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),vimage(nat,nat,suc,F3)))
    <=> pp(aa(set(nat),bool,finite_finite(nat),F3)) ) ).

% finite_vimage_Suc_iff
tff(fact_8081_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),vimage(A,B,F2,A3)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),aa(set(A),set(B),image(A,B,F2),top_top(set(A)))))
       => pp(aa(set(B),bool,finite_finite(B),A3)) ) ) ).

% finite_vimageD'
tff(fact_8082_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(B),A3: 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)),B4),aa(set(A),set(B),image(A,B,F2),A3)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B4)),A3)) ) ) ).

% vimage_subsetI
tff(fact_8083_countable__vimage,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))))
     => ( countable_countable(B,vimage(B,A,F2,B4))
       => countable_countable(A,B4) ) ) ).

% countable_vimage
tff(fact_8084_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(B),A3: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B4)),A3))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% vimage_subset_eq
tff(fact_8085_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( 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)),A3),aa(set(A),set(B),image(A,B,F2),top_top(set(A)))))
       => ( aa(set(A),nat,finite_card(A),vimage(A,B,F2,A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% card_vimage_inj
tff(fact_8086_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),D4: set(A),A3: set(B)] :
      ( inj_on(A,B,F2,D4)
     => ( pp(aa(set(B),bool,finite_finite(B),A3))
       => 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)),inf_inf(set(A)),vimage(A,B,F2,A3)),D4))),aa(set(B),nat,finite_card(B),A3))) ) ) ).

% card_vimage_inj_on_le
tff(fact_8087_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_8088_set__encode__vimage__Suc,axiom,
    ! [A3: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(set(nat),nat,nat_set_encode,A3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% set_encode_vimage_Suc
tff(fact_8089_subset__range__from__nat__into,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(nat),set(A),image(nat,A,counta4804993851260445106t_into(A,A3)),top_top(set(nat))))) ) ).

% subset_range_from_nat_into
tff(fact_8090_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: fun(nat,real),M10: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahv(fun(nat,real),fun(nat,fun(nat,real)),F2),M10),at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,M4))) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_8091_convergent__Suc__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aTP_Lamp_vg(fun(nat,A),fun(nat,A),F2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_Suc_iff
tff(fact_8092_convergent__mult,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [X7: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X7)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X7),Y6)) ) ) ) ).

% convergent_mult
tff(fact_8093_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_vb(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_iff
tff(fact_8094_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_va(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_8095_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_ahx(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_diff_const_right_iff
tff(fact_8096_convergent__diff,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [X7: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X7)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahy(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X7),Y6)) ) ) ) ).

% convergent_diff
tff(fact_8097_lim__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),X: A] :
          ( topolo6863149650580417670ergent(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),topolo3827282254853284352ce_Lim(nat,A,at_top(nat),F2)),X)) ) ) ) ).

% lim_le
tff(fact_8098_Bseq__mono__convergent,axiom,
    ! [X7: fun(nat,real)] :
      ( bfun(nat,real,X7,at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X7,M4)),aa(nat,real,X7,N2))) )
       => topolo6863149650580417670ergent(real,X7) ) ) ).

% Bseq_mono_convergent
tff(fact_8099_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_8100_range__from__nat__into__subset,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,counta4804993851260445106t_into(A,A3)),top_top(set(nat)))),A3)) ) ).

% range_from_nat_into_subset
tff(fact_8101_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: fun(nat,real),M10: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahv(fun(nat,real),fun(nat,fun(nat,real)),F2),M10),at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,M4)),aa(nat,real,F2,N2))) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_8102_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),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,A2),S2)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_8103_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linord329482645794927042rt_key(B,A,F2,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_8104_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),X: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_nb(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_8105_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S2),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,A2),S2)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_8106_cauchy__filter__metric__filtermap,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V768167426530841204y_dist(B)
        & topolo7287701948861334536_space(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( topolo6773858410816713723filter(B,filtermap(A,B,F2,F3))
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [P6: fun(A,bool)] :
                  ( eventually(A,P6,F3)
                  & ! [X5: A,Y5: A] :
                      ( ( pp(aa(A,bool,P6,X5))
                        & pp(aa(A,bool,P6,Y5)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X5),aa(A,B,F2,Y5))),E3)) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_8107_filtermap__at__right__shift,axiom,
    ! [D2: real,A2: real] : filtermap(real,real,aTP_Lamp_ahz(real,fun(real,real),D2),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) = topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),D2),set_ord_greaterThan(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),D2))) ).

% filtermap_at_right_shift
tff(fact_8108_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ) ).

% filtermap_nhds_times
tff(fact_8109_filtermap__nhds__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A2: A] : filtermap(A,A,aTP_Lamp_aia(A,fun(A,A),D2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2)) ) ).

% filtermap_nhds_shift
tff(fact_8110_filtermap__at__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A2: A] : filtermap(A,A,aTP_Lamp_aia(A,fun(A,A),D2),topolo174197925503356063within(A,A2,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))) ) ).

% filtermap_at_shift
tff(fact_8111_filtermap__mono__strong,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(A),G6: filter(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F2,F3)),filtermap(A,B,F2,G6)))
      <=> pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),G6)) ) ) ).

% filtermap_mono_strong
tff(fact_8112_filtermap__inf,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F12: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F23))),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtermap(B,A,F2,F12)),filtermap(B,A,F2,F23)))) ).

% filtermap_inf
tff(fact_8113_filterlim__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
    <=> pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F2,F12)),F23)) ) ).

% filterlim_def
tff(fact_8114_filtercomap__filtermap,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F2: fun(A,B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),filtercomap(A,B,F2,filtermap(A,B,F2,F3)))) ).

% filtercomap_filtermap
tff(fact_8115_filtermap__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),F3: filter(A)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F2,filtercomap(B,A,F2,F3))),F3)) ).

% filtermap_filtercomap
tff(fact_8116_filtermap__le__iff__le__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),F3: filter(B),G6: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F2,F3)),G6))
    <=> pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F3),filtercomap(B,A,F2,G6))) ) ).

% filtermap_le_iff_le_filtercomap
tff(fact_8117_filtermap__mono,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F9: filter(A),F2: fun(A,B)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F3),F9))
     => pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F2,F3)),filtermap(A,B,F2,F9))) ) ).

% filtermap_mono
tff(fact_8118_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F3: fun(C,filter(B)),B4: set(C)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image(C,filter(B),F3),B4)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aib(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F3)),B4)))) ).

% filtermap_INF
tff(fact_8119_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) = filtermap(A,A,aTP_Lamp_aic(A,fun(A,A),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_8120_at__right__to__0,axiom,
    ! [A2: real] : topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)) = filtermap(real,real,aTP_Lamp_aid(real,fun(real,real),A2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% at_right_to_0
tff(fact_8121_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set(A),I5: set(B),F2: fun(D,C),F3: fun(B,filter(D)),G6: fun(A,filter(C))] :
      ( ! [M4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M4),J4))
         => ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
              & pp(aa(filter(C),bool,aa(filter(C),fun(filter(C),bool),ord_less_eq(filter(C)),filtermap(D,C,F2,aa(B,filter(D),F3,X4))),aa(A,filter(C),G6,M4))) ) )
     => filterlim(D,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image(A,filter(C),G6),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image(B,filter(D),F3),I5))) ) ).

% filterlim_INF_INF
tff(fact_8122_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P3: 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,P3,set_ord_greaterThan(A,P3))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P3),set_ord_greaterThan(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P3))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_8123_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_8124_at__right__to__top,axiom,
    topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))) = filtermap(real,real,inverse_inverse(real),at_top(real)) ).

% at_right_to_top
tff(fact_8125_at__top__to__right,axiom,
    at_top(real) = filtermap(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% at_top_to_right
tff(fact_8126_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_8127_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_aie(A,fun(A,bool)),aTP_Lamp_aif(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_8128_span__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_Vector_span(A,B2) = collect(A,aTP_Lamp_aig(set(A),fun(A,bool),B2)) ) ).

% span_explicit
tff(fact_8129_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : real_Vector_span(A,aa(set(A),set(A),insert(A,zero_zero(A)),S)) = real_Vector_span(A,S) ) ).

% span_insert_0
tff(fact_8130_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_8131_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : real_Vector_span(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,zero_zero(A)),bot_bot(set(A))))) = real_Vector_span(A,S) ) ).

% span_delete_0
tff(fact_8132_maximal__independent__subset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V2: set(A)] :
          ~ ! [B8: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V2))
             => ( ~ real_V358717886546972837endent(A,B8)
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B8))) ) ) ) ).

% maximal_independent_subset
tff(fact_8133_spanning__subset__independent,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A3))
         => ( ~ real_V358717886546972837endent(A,A3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),real_Vector_span(A,B4)))
             => ( A3 = B4 ) ) ) ) ) ).

% spanning_subset_independent
tff(fact_8134_maximal__independent__subset__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),V2: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),V2))
         => ( ~ real_V358717886546972837endent(A,S)
           => ~ ! [B8: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),B8))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V2))
                   => ( ~ real_V358717886546972837endent(A,B8)
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B8))) ) ) ) ) ) ) ).

% maximal_independent_subset_extend
tff(fact_8135_span__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T6: set(A)] :
          ( ( real_Vector_span(A,S) = real_Vector_span(A,T6) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),real_Vector_span(A,T6)))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),real_Vector_span(A,S))) ) ) ) ).

% span_eq
tff(fact_8136_span__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),real_Vector_span(A,A3)),real_Vector_span(A,B4))) ) ) ).

% span_mono
tff(fact_8137_span__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),real_Vector_span(A,S))) ) ).

% span_superset
tff(fact_8138_eq__span__insert__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y4: A,S: 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),Y4)),real_Vector_span(A,S)))
         => ( real_Vector_span(A,aa(set(A),set(A),insert(A,X),S)) = real_Vector_span(A,aa(set(A),set(A),insert(A,Y4),S)) ) ) ) ).

% eq_span_insert_eq
tff(fact_8139_span__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S: set(A),Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),real_Vector_span(A,S)))
           => 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),Y4)),real_Vector_span(A,S))) ) ) ) ).

% span_diff
tff(fact_8140_in__span__delete,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A,S: set(A),B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(A,S)))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(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,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,A2),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,B2),bot_bot(set(A)))))))) ) ) ) ).

% in_span_delete
tff(fact_8141_span__breakdown__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A2: A,S: 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,A2),S))))
        <=> ? [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),aa(A,A,real_V8093663219630862766scaleR(A,K3),A2))),real_Vector_span(A,S))) ) ) ).

% span_breakdown_eq
tff(fact_8142_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S: set(A),H: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S)))
         => ( pp(aa(A,bool,H,zero_zero(A)))
           => ( ! [C4: real,X3: A,Y: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                 => ( pp(aa(A,bool,H,Y))
                   => pp(aa(A,bool,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,C4),X3)),Y))) ) )
             => pp(aa(A,bool,H,X)) ) ) ) ) ).

% span_induct_alt
tff(fact_8143_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A2))
       => ( A2 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_8144_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A2 != Top )
      <=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A2),Top)) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_8145_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A2))
      <=> ( A2 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_8146_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A2)) ) ).

% ordering_top.extremum_strict
tff(fact_8147_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),Top)) ) ).

% ordering_top.extremum
tff(fact_8148_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_span(A,S))) ) ).

% span_0
tff(fact_8149_span__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A,S: set(A)] : real_Vector_span(A,aa(set(A),set(A),insert(A,A2),S)) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_aih(A,fun(set(A),fun(A,bool)),A2),S)) ) ).

% span_insert
tff(fact_8150_dependent__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [P: set(A)] :
          ( real_V358717886546972837endent(A,P)
        <=> ? [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),P))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),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,X5),bot_bot(set(A))))))) ) ) ) ).

% dependent_def
tff(fact_8151_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_mz(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_8152_span__image__scale,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),C2: fun(A,real)] :
          ( pp(aa(set(A),bool,finite_finite(A),S))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => ( aa(A,real,C2,X3) != zero_zero(real) ) )
           => ( real_Vector_span(A,aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(fun(A,real),fun(A,A),C2)),S)) = real_Vector_span(A,S) ) ) ) ) ).

% span_image_scale
tff(fact_8153_span__breakdown,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,S: set(A),A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),S))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(A,S)))
           => ? [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),A2),aa(A,A,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)),S),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))) ) ) ) ).

% span_breakdown
tff(fact_8154_independent__span__bound,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T6: set(A),S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),T6))
         => ( ~ real_V358717886546972837endent(A,S)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),real_Vector_span(A,T6)))
             => ( pp(aa(set(A),bool,finite_finite(A),S))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),S)),aa(set(A),nat,finite_card(A),T6))) ) ) ) ) ) ).

% independent_span_bound
tff(fact_8155_exchange__lemma,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T6: set(A),S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),T6))
         => ( ~ real_V358717886546972837endent(A,S)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),real_Vector_span(A,T6)))
             => ? [T13: set(A)] :
                  ( ( aa(set(A),nat,finite_card(A),T13) = aa(set(A),nat,finite_card(A),T6) )
                  & pp(aa(set(A),bool,finite_finite(A),T13))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T13))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T13),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T6)))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),real_Vector_span(A,T13))) ) ) ) ) ) ).

% exchange_lemma
tff(fact_8156_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_8157_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_kp(nat,fun(nat,bool)),aTP_Lamp_ec(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_8158_span__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A)] : real_Vector_span(A,B4) = collect(A,aTP_Lamp_aii(set(A),fun(A,bool),B4)) ) ).

% span_alt
tff(fact_8159_span__explicit_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_Vector_span(A,B2) = collect(A,aTP_Lamp_aij(set(A),fun(A,bool),B2)) ) ).

% span_explicit'
tff(fact_8160_representation__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A] :
          ( ( ( ~ real_V358717886546972837endent(A,Basis)
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis))) )
           => ( real_V7696804695334737415tation(A,Basis,V) = fChoice(fun(A,real),aa(A,fun(fun(A,real),bool),aTP_Lamp_aik(set(A),fun(A,fun(fun(A,real),bool)),Basis),V)) ) )
          & ( ~ ( ~ real_V358717886546972837endent(A,Basis)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis))) )
           => ( real_V7696804695334737415tation(A,Basis,V) = aTP_Lamp_ail(A,real) ) ) ) ) ).

% representation_def
tff(fact_8161_extend__basis__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A)] : real_V4986007116245087402_basis(A,B4) = fChoice(set(A),aTP_Lamp_aim(set(A),fun(set(A),bool),B4)) ) ).

% extend_basis_def
tff(fact_8162_finite__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_ain(set(A),fun(A,fun(A,bool)),Basis),V)))) ) ).

% finite_representation
tff(fact_8163_representation__ne__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,B2: A] :
          ( ( aa(A,real,real_V7696804695334737415tation(A,Basis,V),B2) != zero_zero(real) )
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),Basis)) ) ) ).

% representation_ne_zero
tff(fact_8164_representation__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),X4: A] : aa(A,real,real_V7696804695334737415tation(A,Basis,zero_zero(A)),X4) = zero_zero(real) ) ).

% representation_zero
tff(fact_8165_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))
           => ! [X4: A] :
                ( ( ( X4 = B2 )
                 => ( aa(A,real,real_V7696804695334737415tation(A,Basis,B2),X4) = one_one(real) ) )
                & ( ( X4 != B2 )
                 => ( aa(A,real,real_V7696804695334737415tation(A,Basis,B2),X4) = zero_zero(real) ) ) ) ) ) ) ).

% representation_basis
tff(fact_8166_representation__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,Basis2: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis2)))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Basis2),Basis))
             => ( real_V7696804695334737415tation(A,Basis,V) = real_V7696804695334737415tation(A,Basis2,V) ) ) ) ) ) ).

% representation_extend
tff(fact_8167_extend__basis__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A)] :
          ( ~ real_V358717886546972837endent(A,B4)
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),real_V4986007116245087402_basis(A,B4))) ) ) ).

% extend_basis_superset
tff(fact_8168_representation__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,U: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis)))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),real_Vector_span(A,Basis)))
             => ! [X4: A] : aa(A,real,real_V7696804695334737415tation(A,Basis,aa(A,A,aa(A,fun(A,A),minus_minus(A),U),V)),X4) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(A,real,real_V7696804695334737415tation(A,Basis,U),X4)),aa(A,real,real_V7696804695334737415tation(A,Basis,V),X4)) ) ) ) ) ).

% representation_diff
tff(fact_8169_representation__scale,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,R2: real] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis)))
           => ! [X4: A] : aa(A,real,real_V7696804695334737415tation(A,Basis,aa(A,A,real_V8093663219630862766scaleR(A,R2),V)),X4) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(A,real,real_V7696804695334737415tation(A,Basis,V),X4)) ) ) ) ).

% representation_scale
tff(fact_8170_sum__nonzero__representation__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis)))
           => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aa(A,fun(A,A),aTP_Lamp_aio(set(A),fun(A,fun(A,A)),Basis),V)),collect(A,aa(A,fun(A,bool),aTP_Lamp_ain(set(A),fun(A,fun(A,bool)),Basis),V))) = V ) ) ) ) ).

% sum_nonzero_representation_eq
tff(fact_8171_sum__representation__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,B4: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis)))
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Basis),B4))
               => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aa(A,fun(A,A),aTP_Lamp_aio(set(A),fun(A,fun(A,A)),Basis),V)),B4) = V ) ) ) ) ) ) ).

% sum_representation_eq
tff(fact_8172_representation__eqI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V: A,F2: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V),real_Vector_span(A,Basis)))
           => ( ! [B3: A] :
                  ( ( aa(A,real,F2,B3) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),Basis)) )
             => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F2))))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),F2)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F2))) = V )
                 => ( real_V7696804695334737415tation(A,Basis,V) = F2 ) ) ) ) ) ) ) ).

% representation_eqI
tff(fact_8173_construct__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [B4: set(A),G: fun(A,B),V: A] : real_V4425403222259421789struct(A,B,B4,G,V) = 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_aip(set(A),fun(fun(A,B),fun(A,fun(A,B))),B4),G),V)),collect(A,aa(A,fun(A,bool),aTP_Lamp_aiq(set(A),fun(A,fun(A,bool)),B4),V))) ) ).

% construct_def
tff(fact_8174_dim__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V2: set(A)] :
          ( ( ? [B9: set(A)] :
                ( ~ real_V358717886546972837endent(A,B9)
                & ( real_Vector_span(A,B9) = real_Vector_span(A,V2) ) )
           => ( real_Vector_dim(A,V2) = aa(set(A),nat,finite_card(A),fChoice(set(A),aTP_Lamp_air(set(A),fun(set(A),bool),V2))) ) )
          & ( ~ ? [B3: set(A)] :
                  ( ~ real_V358717886546972837endent(A,B3)
                  & ( real_Vector_span(A,B3) = real_Vector_span(A,V2) ) )
           => ( real_Vector_dim(A,V2) = zero_zero(nat) ) ) ) ) ).

% dim_def
tff(fact_8175_dim__le__card_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,S2)),aa(set(A),nat,finite_card(A),S2))) ) ) ).

% dim_le_card'
tff(fact_8176_dim__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),V2: set(A),N: nat] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),V2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B4)))
           => ( ~ real_V358717886546972837endent(A,B4)
             => ( ( aa(set(A),nat,finite_card(A),B4) = N )
               => ( real_Vector_dim(A,V2) = N ) ) ) ) ) ) ).

% dim_unique
tff(fact_8177_basis__exists,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V2: set(A)] :
          ~ ! [B8: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V2))
             => ( ~ real_V358717886546972837endent(A,B8)
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B8)))
                 => ( aa(set(A),nat,finite_card(A),B8) != real_Vector_dim(A,V2) ) ) ) ) ) ).

% basis_exists
tff(fact_8178_basis__card__eq__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),V2: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),V2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B4)))
           => ( ~ real_V358717886546972837endent(A,B4)
             => ( aa(set(A),nat,finite_card(A),B4) = real_Vector_dim(A,V2) ) ) ) ) ) ).

% basis_card_eq_dim
tff(fact_8179_span__card__ge__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B4: set(A),V2: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),V2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,B4)))
           => ( pp(aa(set(A),bool,finite_finite(A),B4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V2)),aa(set(A),nat,finite_card(A),B4))) ) ) ) ) ).

% span_card_ge_dim
tff(fact_8180_dim__le__card,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V2: set(A),W3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V2),real_Vector_span(A,W3)))
         => ( pp(aa(set(A),bool,finite_finite(A),W3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V2)),aa(set(A),nat,finite_card(A),W3))) ) ) ) ).

% dim_le_card
tff(fact_8181_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_aea(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_8182_ATP_Olambda__2,axiom,
    ! [Uu: fun(nat,rat)] : aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ago(fun(nat,rat),fun(nat,rat)),Uu) = if(fun(nat,rat),vanishes(Uu),aTP_Lamp_agm(nat,rat),aTP_Lamp_agn(fun(nat,rat),fun(nat,rat),Uu)) ).

% ATP.lambda_2
tff(fact_8183_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_as(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_8184_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_up(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_8185_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_mk(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_8186_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_abv(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_8187_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_cc(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_8188_ATP_Olambda__8,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_hx(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))))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_8
tff(fact_8189_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_adt(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_8190_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vz(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_8191_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_fe(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_11
tff(fact_8192_ATP_Olambda__12,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_wa(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_12
tff(fact_8193_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ut(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_13
tff(fact_8194_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_yc(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_14
tff(fact_8195_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_np(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_15
tff(fact_8196_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vl(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_16
tff(fact_8197_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_mg(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_17
tff(fact_8198_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cn(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_18
tff(fact_8199_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_afz(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uu),zero_zero(nat)) ).

% ATP.lambda_19
tff(fact_8200_ATP_Olambda__20,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_ads(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_20
tff(fact_8201_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vt(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_21
tff(fact_8202_ATP_Olambda__22,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vy(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_22
tff(fact_8203_ATP_Olambda__23,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afb(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_23
tff(fact_8204_ATP_Olambda__24,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_afc(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_24
tff(fact_8205_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_afd(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_25
tff(fact_8206_ATP_Olambda__26,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_qq(real,real),Uu) = suminf(real,aTP_Lamp_at(real,fun(nat,real),Uu)) ).

% ATP.lambda_26
tff(fact_8207_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_aem(nat,set(nat)),Uu) = collect(nat,aTP_Lamp_kf(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_27
tff(fact_8208_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real] : aa(real,filter(A),aTP_Lamp_yh(real,filter(A)),Uu) = principal(A,collect(A,aTP_Lamp_yg(real,fun(A,bool),Uu))) ) ).

% ATP.lambda_28
tff(fact_8209_ATP_Olambda__29,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_abs(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),collect(product_prod(complex,complex),product_case_prod(complex,complex,bool,aTP_Lamp_abr(real,fun(complex,fun(complex,bool)),Uu)))) ).

% ATP.lambda_29
tff(fact_8210_ATP_Olambda__30,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_abq(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),collect(product_prod(real,real),product_case_prod(real,real,bool,aTP_Lamp_abp(real,fun(real,fun(real,bool)),Uu)))) ).

% ATP.lambda_30
tff(fact_8211_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_abo(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),collect(product_prod(A,A),product_case_prod(A,A,bool,aTP_Lamp_abn(real,fun(A,fun(A,bool)),Uu)))) ) ).

% ATP.lambda_31
tff(fact_8212_ATP_Olambda__32,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vv(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_32
tff(fact_8213_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vn(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_33
tff(fact_8214_ATP_Olambda__34,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afa(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_34
tff(fact_8215_ATP_Olambda__35,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_aes(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_35
tff(fact_8216_ATP_Olambda__36,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_abj(A,B),Uu) = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,Uu)) ) ).

% ATP.lambda_36
tff(fact_8217_ATP_Olambda__37,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_abk(A,B),Uu) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu)) ) ).

% ATP.lambda_37
tff(fact_8218_ATP_Olambda__38,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_adp(num,option(num)),Uu) = some(num,aa(num,num,bit1,Uu)) ).

% ATP.lambda_38
tff(fact_8219_ATP_Olambda__39,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ado(num,option(num)),Uu) = some(num,aa(num,num,bit0,Uu)) ).

% ATP.lambda_39
tff(fact_8220_ATP_Olambda__40,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hn(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_40
tff(fact_8221_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_acw(A,bool),Uu))
        <=> ? [N4: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N4) )
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N4)) ) ) ) ).

% ATP.lambda_41
tff(fact_8222_ATP_Olambda__42,axiom,
    ! [Uu: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aTP_Lamp_agt(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] :
            ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,Uu,N4))) ) ) ) ).

% ATP.lambda_42
tff(fact_8223_ATP_Olambda__43,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_acz(real,bool),Uu))
    <=> ? [I2: int,N4: 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),N4)) )
          & ( N4 != zero_zero(nat) ) ) ) ).

% ATP.lambda_43
tff(fact_8224_ATP_Olambda__44,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_ada(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_44
tff(fact_8225_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_acj(product_prod(A,A),bool),Uu))
        <=> ? [X5: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X5),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y5)) ) ) ) ).

% ATP.lambda_45
tff(fact_8226_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ack(product_prod(A,A),bool),Uu))
        <=> ? [X5: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X5),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),X5)) ) ) ) ).

% ATP.lambda_46
tff(fact_8227_ATP_Olambda__47,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_adk(nat,option(num)),Uu) = some(num,one2) ).

% ATP.lambda_47
tff(fact_8228_ATP_Olambda__48,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_adw(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)),some(num,one2)),aTP_Lamp_adu(nat,fun(num,option(num)),Uua)),aTP_Lamp_adv(nat,fun(num,option(num)),Uua)),Uu) ).

% ATP.lambda_48
tff(fact_8229_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cf(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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_49
tff(fact_8230_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dz(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_50
tff(fact_8231_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cg(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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_51
tff(fact_8232_ATP_Olambda__52,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_cd(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_52
tff(fact_8233_ATP_Olambda__53,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_nc(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))),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))),code_int_of_integer(Uu))),one_one(int))) ).

% ATP.lambda_53
tff(fact_8234_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hz(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_54
tff(fact_8235_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_aec(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_55
tff(fact_8236_ATP_Olambda__56,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_na(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_56
tff(fact_8237_ATP_Olambda__57,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ja(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_57
tff(fact_8238_ATP_Olambda__58,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_afx(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_58
tff(fact_8239_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ea(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_59
tff(fact_8240_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_afm(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_60
tff(fact_8241_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_adx(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_adw(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_61
tff(fact_8242_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_adu(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ado(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_62
tff(fact_8243_ATP_Olambda__63,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_adq(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ado(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_63
tff(fact_8244_ATP_Olambda__64,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aer(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aeq(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_64
tff(fact_8245_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_65
tff(fact_8246_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hu(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_ht(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_66
tff(fact_8247_ATP_Olambda__67,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_adz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = 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(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_67
tff(fact_8248_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ao(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_68
tff(fact_8249_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bv(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_69
tff(fact_8250_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ch(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_70
tff(fact_8251_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_du(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_71
tff(fact_8252_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dv(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_72
tff(fact_8253_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hy(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)))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_73
tff(fact_8254_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_ho(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_74
tff(fact_8255_ATP_Olambda__75,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_co(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_75
tff(fact_8256_ATP_Olambda__76,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aeb(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_76
tff(fact_8257_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_iy(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_77
tff(fact_8258_ATP_Olambda__78,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_iz(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_78
tff(fact_8259_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_at(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_79
tff(fact_8260_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qr(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_80
tff(fact_8261_ATP_Olambda__81,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uz(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_81
tff(fact_8262_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dy(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_82
tff(fact_8263_ATP_Olambda__83,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_afk(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_83
tff(fact_8264_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cz(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_84
tff(fact_8265_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,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_85
tff(fact_8266_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_aim(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uu),Uua))
            & ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = top_top(set(A)) ) ) ) ) ).

% ATP.lambda_86
tff(fact_8267_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dj(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_87
tff(fact_8268_ATP_Olambda__88,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mz(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_88
tff(fact_8269_ATP_Olambda__89,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_ml(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_89
tff(fact_8270_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_do(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_90
tff(fact_8271_ATP_Olambda__91,axiom,
    ! [Uu: set(int),Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_acu(set(int),fun(int,bool),Uu),Uua))
    <=> ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Uua),Uu))
        & ! [X5: int] :
            ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X5),Uu))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),Uua)) ) ) ) ).

% ATP.lambda_91
tff(fact_8272_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_ahp(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( 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))),Uu),Uua))
        & ! [A6: A,B7: A,C6: 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,A6),B7)),Uua))
              & 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,B7),C6)),Uu)) )
           => 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,A6),B7)),Uu)) ) ) ) ).

% ATP.lambda_92
tff(fact_8273_ATP_Olambda__93,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ft(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_93
tff(fact_8274_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_ye(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_94
tff(fact_8275_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_aho(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))
        & chain_subset(A,Uua) ) ) ).

% ATP.lambda_95
tff(fact_8276_ATP_Olambda__96,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_pd(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu))
        & pp(aa(set(A),bool,finite_finite(A),Uua)) ) ) ).

% ATP.lambda_96
tff(fact_8277_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agi(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_97
tff(fact_8278_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afn(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_98
tff(fact_8279_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ct(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_99
tff(fact_8280_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_av(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_100
tff(fact_8281_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ki(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_101
tff(fact_8282_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_kh(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_102
tff(fact_8283_ATP_Olambda__103,axiom,
    ! [D: $tType,B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(D,product_prod(A,B))] :
      ( pp(aa(fun(D,product_prod(A,B)),bool,aTP_Lamp_afw(fun(A,fun(B,bool)),fun(fun(D,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(set(D),set(product_prod(A,B)),image(D,product_prod(A,B),Uua),top_top(set(D)))),collect(product_prod(A,B),product_case_prod(A,B,bool,Uu)))) ) ).

% ATP.lambda_103
tff(fact_8284_ATP_Olambda__104,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_wf(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_104
tff(fact_8285_ATP_Olambda__105,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_xm(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_105
tff(fact_8286_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_yf(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),Uua)),Uua) ).

% ATP.lambda_106
tff(fact_8287_ATP_Olambda__107,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_107
tff(fact_8288_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hw(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))
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_108
tff(fact_8289_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aha(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_109
tff(fact_8290_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ahc(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_110
tff(fact_8291_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hf(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_111
tff(fact_8292_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fb(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_112
tff(fact_8293_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vp(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_113
tff(fact_8294_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_114
tff(fact_8295_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_115
tff(fact_8296_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_id(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_116
tff(fact_8297_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bj(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_117
tff(fact_8298_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ap(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_118
tff(fact_8299_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jt(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_119
tff(fact_8300_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bi(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_120
tff(fact_8301_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gf(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_121
tff(fact_8302_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ge(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_122
tff(fact_8303_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vq(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_123
tff(fact_8304_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ez(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_124
tff(fact_8305_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahl(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5)) ) ) ) ) ).

% ATP.lambda_125
tff(fact_8306_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aaj(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),Uua)) ) ) ) ) ).

% ATP.lambda_126
tff(fact_8307_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gy(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_127
tff(fact_8308_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ga(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_128
tff(fact_8309_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_ahe(fun(A,real),fun(A,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua)),Uua) ) ).

% ATP.lambda_129
tff(fact_8310_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yn(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real))) ) ).

% ATP.lambda_130
tff(fact_8311_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_131
tff(fact_8312_ATP_Olambda__132,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_en(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_132
tff(fact_8313_ATP_Olambda__133,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_ke(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_133
tff(fact_8314_ATP_Olambda__134,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_kg(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_134
tff(fact_8315_ATP_Olambda__135,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wp(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_uz(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_135
tff(fact_8316_ATP_Olambda__136,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wo(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_uz(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_136
tff(fact_8317_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_aeo(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_aen(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_137
tff(fact_8318_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_nq(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_138
tff(fact_8319_ATP_Olambda__139,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_wy(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_139
tff(fact_8320_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ar(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_140
tff(fact_8321_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aq(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_141
tff(fact_8322_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_air(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_142
tff(fact_8323_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_br(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_143
tff(fact_8324_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bw(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_144
tff(fact_8325_ATP_Olambda__145,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_adj(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_145
tff(fact_8326_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_pe(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_146
tff(fact_8327_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_wk(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_147
tff(fact_8328_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_wj(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_148
tff(fact_8329_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fd(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_149
tff(fact_8330_ATP_Olambda__150,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fc(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_150
tff(fact_8331_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_xx(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_151
tff(fact_8332_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aeg(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,finite_finite(A),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_152
tff(fact_8333_ATP_Olambda__153,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
      ( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_ahd(fun(A,fun(B,bool)),fun(list(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(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),collect(product_prod(A,B),product_case_prod(A,B,bool,Uu)))) ) ).

% ATP.lambda_153
tff(fact_8334_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ng(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_154
tff(fact_8335_ATP_Olambda__155,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_afy(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_155
tff(fact_8336_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gg(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_156
tff(fact_8337_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_aff(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_157
tff(fact_8338_ATP_Olambda__158,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_afh(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_158
tff(fact_8339_ATP_Olambda__159,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_go(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_159
tff(fact_8340_ATP_Olambda__160,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_gr(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),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_160
tff(fact_8341_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gn(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),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_161
tff(fact_8342_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dx(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_162
tff(fact_8343_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_mb(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_163
tff(fact_8344_ATP_Olambda__164,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_wh(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_164
tff(fact_8345_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wd(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_165
tff(fact_8346_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ij(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_166
tff(fact_8347_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_agq(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_167
tff(fact_8348_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_we(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_168
tff(fact_8349_ATP_Olambda__169,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,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_169
tff(fact_8350_ATP_Olambda__170,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_pc(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),Uu),set_ord_lessThan(nat,Uua)) ).

% ATP.lambda_170
tff(fact_8351_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ox(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_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_171
tff(fact_8352_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_yg(real,fun(A,bool),Uu),Uua))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uu),real_V7770717601297561774m_norm(A,Uua))) ) ) ).

% ATP.lambda_172
tff(fact_8353_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ahb(set(product_prod(A,A)),fun(nat,bool),Uu),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_173
tff(fact_8354_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_vm(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_174
tff(fact_8355_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aet(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_175
tff(fact_8356_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hi(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_176
tff(fact_8357_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_im(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_177
tff(fact_8358_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hd(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_178
tff(fact_8359_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_nr(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_179
tff(fact_8360_ATP_Olambda__180,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_kq(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).

% ATP.lambda_180
tff(fact_8361_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kp(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_181
tff(fact_8362_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adg(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_182
tff(fact_8363_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aie(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_183
tff(fact_8364_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ym(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_184
tff(fact_8365_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ew(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_185
tff(fact_8366_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pb(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_186
tff(fact_8367_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ws(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_8368_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_8369_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_abw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8370_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ec(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_190
tff(fact_8371_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adh(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_191
tff(fact_8372_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aif(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_192
tff(fact_8373_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_193
tff(fact_8374_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_8375_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ai(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8376_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oe(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_8377_ATP_Olambda__197,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_ahz(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uu) ).

% ATP.lambda_197
tff(fact_8378_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ns(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_198
tff(fact_8379_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aia(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_8380_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_8381_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_on(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_8382_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_pw(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_202
tff(fact_8383_ATP_Olambda__203,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aid(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uua),Uu) ).

% ATP.lambda_203
tff(fact_8384_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_me(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_204
tff(fact_8385_ATP_Olambda__205,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_os(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_205
tff(fact_8386_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aic(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_8387_ATP_Olambda__207,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_pz(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_207
tff(fact_8388_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_kf(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_208
tff(fact_8389_ATP_Olambda__209,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_kv(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_209
tff(fact_8390_ATP_Olambda__210,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_210
tff(fact_8391_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_nf(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uua),Uu) ).

% ATP.lambda_211
tff(fact_8392_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_cx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_212
tff(fact_8393_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_aen(list(A),fun(A,list(A))),Uu),Uua) = cons(A,Uua,Uu) ).

% ATP.lambda_213
tff(fact_8394_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_acb(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_214
tff(fact_8395_ATP_Olambda__215,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_vu(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_215
tff(fact_8396_ATP_Olambda__216,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aak(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uua),Uu)) ) ).

% ATP.lambda_216
tff(fact_8397_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_abu(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_217
tff(fact_8398_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahh(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_218
tff(fact_8399_ATP_Olambda__219,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahk(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_219
tff(fact_8400_ATP_Olambda__220,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_220
tff(fact_8401_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_agz(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_221
tff(fact_8402_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_aeu(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_222
tff(fact_8403_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_lu(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_223
tff(fact_8404_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_zt(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_224
tff(fact_8405_ATP_Olambda__225,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_zm(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_225
tff(fact_8406_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_zh(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_226
tff(fact_8407_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yl(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_227
tff(fact_8408_ATP_Olambda__228,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_agj(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_228
tff(fact_8409_ATP_Olambda__229,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fs(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_229
tff(fact_8410_ATP_Olambda__230,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fr(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_230
tff(fact_8411_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_zx(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_231
tff(fact_8412_ATP_Olambda__232,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_afi(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_232
tff(fact_8413_ATP_Olambda__233,axiom,
    ! [Uu: fun(real,bool),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_zl(fun(real,bool),fun(real,bool),Uu),Uua))
    <=> pp(aa(real,bool,Uu,aa(real,real,inverse_inverse(real),Uua))) ) ).

% ATP.lambda_233
tff(fact_8414_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_yd(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_234
tff(fact_8415_ATP_Olambda__235,axiom,
    ! [Uu: fun(real,real),Uua: nat] : aa(nat,real,aTP_Lamp_afo(fun(real,real),fun(nat,real),Uu),Uua) = aa(real,real,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ).

% ATP.lambda_235
tff(fact_8416_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_zv(fun(real,A),fun(nat,A),Uu),Uua) = aa(real,A,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_236
tff(fact_8417_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_aae(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_237
tff(fact_8418_ATP_Olambda__238,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_afj(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_238
tff(fact_8419_ATP_Olambda__239,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aaa(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_239
tff(fact_8420_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_240
tff(fact_8421_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_241
tff(fact_8422_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_242
tff(fact_8423_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_243
tff(fact_8424_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ey(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_244
tff(fact_8425_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_245
tff(fact_8426_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_adv(nat,fun(num,option(num)),Uu),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_246
tff(fact_8427_ATP_Olambda__247,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_adr(num,fun(nat,option(num)),Uu),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_247
tff(fact_8428_ATP_Olambda__248,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_agv(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_agu(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_248
tff(fact_8429_ATP_Olambda__249,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_no(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_nn(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_249
tff(fact_8430_ATP_Olambda__250,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_nm(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_nl(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_250
tff(fact_8431_ATP_Olambda__251,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_nk(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_nj(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_251
tff(fact_8432_ATP_Olambda__252,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_ni(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_nh(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_252
tff(fact_8433_ATP_Olambda__253,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_ne(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_nd(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_253
tff(fact_8434_ATP_Olambda__254,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_qi(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_qh(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_254
tff(fact_8435_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_px(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_255
tff(fact_8436_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_xl(real,fun(A,set(A)),Uu),Uua) = collect(A,aa(A,fun(A,bool),aTP_Lamp_xk(real,fun(A,fun(A,bool)),Uu),Uua)) ) ).

% ATP.lambda_256
tff(fact_8437_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_fy(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_257
tff(fact_8438_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_rp(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_258
tff(fact_8439_ATP_Olambda__259,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_yi(fun(real,real),fun(real,bool),Uu),Uua))
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_259
tff(fact_8440_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Uu),Uua))
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_260
tff(fact_8441_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aj(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_261
tff(fact_8442_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_pa(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_262
tff(fact_8443_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( field_char_0(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),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_263
tff(fact_8444_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ak(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_264
tff(fact_8445_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agh(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_265
tff(fact_8446_ATP_Olambda__266,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_wc(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_266
tff(fact_8447_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aei(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_267
tff(fact_8448_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aeh(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_268
tff(fact_8449_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ix(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_269
tff(fact_8450_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_iw(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_270
tff(fact_8451_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ad(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_271
tff(fact_8452_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aey(A,fun(A,bool),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_272
tff(fact_8453_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bo(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8454_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_dd(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8455_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bq(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8456_ATP_Olambda__276,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_cv(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8457_ATP_Olambda__277,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_gu(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8458_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_th(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8459_ATP_Olambda__279,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_mp(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8460_ATP_Olambda__280,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_vs(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_280
tff(fact_8461_ATP_Olambda__281,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aTP_Lamp_agn(fun(nat,rat),fun(nat,rat),Uu),Uua) = aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_281
tff(fact_8462_ATP_Olambda__282,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_rn(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8463_ATP_Olambda__283,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_tk(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8464_ATP_Olambda__284,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aap(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8465_ATP_Olambda__285,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aba(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8466_ATP_Olambda__286,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_286
tff(fact_8467_ATP_Olambda__287,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_um(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8468_ATP_Olambda__288,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8469_ATP_Olambda__289,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8470_ATP_Olambda__290,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_em(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_290
tff(fact_8471_ATP_Olambda__291,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_gp(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8472_ATP_Olambda__292,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cr(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8473_ATP_Olambda__293,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_aa(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_293
tff(fact_8474_ATP_Olambda__294,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,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8475_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8476_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8477_ATP_Olambda__297,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_kn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_297
tff(fact_8478_ATP_Olambda__298,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_afv(fun(nat,rat),fun(nat,rat)),Uu),Uua) = aa(rat,rat,uminus_uminus(rat),aa(nat,rat,Uu,Uua)) ).

% ATP.lambda_298
tff(fact_8479_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hb(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_299
tff(fact_8480_ATP_Olambda__300,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_gq(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8481_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_cs(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8482_ATP_Olambda__302,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gd(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_302
tff(fact_8483_ATP_Olambda__303,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_si(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8484_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8485_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abi(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8486_ATP_Olambda__306,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_abe(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8487_ATP_Olambda__307,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_zn(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arcosh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_307
tff(fact_8488_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zi(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8489_ATP_Olambda__309,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qu(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_8490_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8491_ATP_Olambda__311,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_tj(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,sgn_sgn(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8492_ATP_Olambda__312,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aaz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8493_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_un(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8494_ATP_Olambda__314,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bp(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_314
tff(fact_8495_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_315
tff(fact_8496_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_eh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8497_ATP_Olambda__317,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [Uu: fun(A10,A10),Uua: A10] : aa(A10,A10,aTP_Lamp_qf(fun(A10,A10),fun(A10,A10),Uu),Uua) = aa(A10,A10,tanh(A10),aa(A10,A10,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8498_ATP_Olambda__318,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_abd(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_8499_ATP_Olambda__319,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_ur(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8500_ATP_Olambda__320,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_tm(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_320
tff(fact_8501_ATP_Olambda__321,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [Uu: fun(A10,A10),Uua: A10] : aa(A10,A10,aTP_Lamp_pm(fun(A10,A10),fun(A10,A10),Uu),Uua) = sinh(A10,aa(A10,A10,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_8502_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rk(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_322
tff(fact_8503_ATP_Olambda__323,axiom,
    ! [A10: $tType] :
      ( ( real_Vector_banach(A10)
        & real_V3459762299906320749_field(A10) )
     => ! [Uu: fun(A10,A10),Uua: A10] : aa(A10,A10,aTP_Lamp_pn(fun(A10,A10),fun(A10,A10),Uu),Uua) = cosh(A10,aa(A10,A10,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8504_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rj(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_8505_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_325
tff(fact_8506_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_tl(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8507_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rh(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8508_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pr(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8509_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8510_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ps(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8511_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_tn(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_8512_ATP_Olambda__332,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wn(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_332
tff(fact_8513_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rs(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8514_ATP_Olambda__334,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pt(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8515_ATP_Olambda__335,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_xs(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_335
tff(fact_8516_ATP_Olambda__336,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_xt(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_336
tff(fact_8517_ATP_Olambda__337,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_ot(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_337
tff(fact_8518_ATP_Olambda__338,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_338
tff(fact_8519_ATP_Olambda__339,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_us(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8520_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_340
tff(fact_8521_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aax(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8522_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vd(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8523_ATP_Olambda__343,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_343
tff(fact_8524_ATP_Olambda__344,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_oo(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_344
tff(fact_8525_ATP_Olambda__345,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ll(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_345
tff(fact_8526_ATP_Olambda__346,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_adc(fun(B,bool),fun(B,bool),Uu),Uua))
    <=> ~ pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_346
tff(fact_8527_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ae(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8528_ATP_Olambda__348,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_aef(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),collect(set(A),aa(set(A),fun(set(A),bool),aTP_Lamp_aee(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_348
tff(fact_8529_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_xv(A,fun(real,filter(A)),Uu),Uua) = principal(A,collect(A,aa(real,fun(A,bool),aTP_Lamp_xu(A,fun(real,fun(A,bool)),Uu),Uua))) ) ).

% ATP.lambda_349
tff(fact_8530_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_uv(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_350
tff(fact_8531_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aii(set(A),fun(A,bool),Uu),Uua))
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),F7)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F7))) )
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F7))),Uu))
              & pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F7)))) ) ) ) ).

% ATP.lambda_351
tff(fact_8532_ATP_Olambda__352,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aij(set(A),fun(A,bool),Uu),Uua))
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),F7)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F7))) )
              & pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),F7))))
              & ! [V6: A] :
                  ( ( aa(A,real,F7,V6) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),Uu)) ) ) ) ) ).

% ATP.lambda_352
tff(fact_8533_ATP_Olambda__353,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_acr(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu) )
              & ! [X5: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).

% ATP.lambda_353
tff(fact_8534_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_acs(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu) )
              & ! [X5: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).

% ATP.lambda_354
tff(fact_8535_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_acx(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu) )
              & ! [X5: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X5),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).

% ATP.lambda_355
tff(fact_8536_ATP_Olambda__356,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_act(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B10: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B10) )
          & pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Uu),pow2(A,B10))) ) ) ).

% ATP.lambda_356
tff(fact_8537_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( pp(aa(filter(A),bool,aTP_Lamp_acv(set(filter(A)),fun(filter(A),bool),Uu),Uua))
    <=> ! [X5: filter(A)] :
          ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X5),Uu))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),Uua),X5)) ) ) ).

% ATP.lambda_357
tff(fact_8538_ATP_Olambda__358,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_acp(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X5)) ) ) ) ).

% ATP.lambda_358
tff(fact_8539_ATP_Olambda__359,axiom,
    ! [Uu: set(real),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_ahi(set(real),fun(real,bool),Uu),Uua))
    <=> ! [X5: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X5),Uu))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X5),Uua)) ) ) ).

% ATP.lambda_359
tff(fact_8540_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aco(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Uua)) ) ) ) ).

% ATP.lambda_360
tff(fact_8541_ATP_Olambda__361,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aai(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5))
             => pp(aa(A,bool,Uu,Y5)) ) ) ) ).

% ATP.lambda_361
tff(fact_8542_ATP_Olambda__362,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_acl(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X5: B] : Uua = aa(B,A,Uu,X5) ) ).

% ATP.lambda_362
tff(fact_8543_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aig(set(A),fun(A,bool),Uu),Uua))
        <=> ? [T3: set(A),R5: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),R5)),T3) )
              & pp(aa(set(A),bool,finite_finite(A),T3))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),Uu)) ) ) ) ).

% ATP.lambda_363
tff(fact_8544_ATP_Olambda__364,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agg(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,Uu) ).

% ATP.lambda_364
tff(fact_8545_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aTP_Lamp_agx(nat,fun(nat,rat),Uu),Uua) = aa(nat,rat,semiring_1_of_nat(rat),Uu) ).

% ATP.lambda_365
tff(fact_8546_ATP_Olambda__366,axiom,
    ! [Uu: int,Uua: nat] : aa(nat,rat,aTP_Lamp_agw(int,fun(nat,rat),Uu),Uua) = aa(int,rat,ring_1_of_int(rat),Uu) ).

% ATP.lambda_366
tff(fact_8547_ATP_Olambda__367,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ce(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_367
tff(fact_8548_ATP_Olambda__368,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fq(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_368
tff(fact_8549_ATP_Olambda__369,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_is(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_369
tff(fact_8550_ATP_Olambda__370,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hj(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_370
tff(fact_8551_ATP_Olambda__371,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hk(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_371
tff(fact_8552_ATP_Olambda__372,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_hm(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_372
tff(fact_8553_ATP_Olambda__373,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_js(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_373
tff(fact_8554_ATP_Olambda__374,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_ms(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_374
tff(fact_8555_ATP_Olambda__375,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_jq(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_375
tff(fact_8556_ATP_Olambda__376,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_ax(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_376
tff(fact_8557_ATP_Olambda__377,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_jp(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_377
tff(fact_8558_ATP_Olambda__378,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_kz(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_378
tff(fact_8559_ATP_Olambda__379,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_mm(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_379
tff(fact_8560_ATP_Olambda__380,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_ky(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_380
tff(fact_8561_ATP_Olambda__381,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_jr(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_381
tff(fact_8562_ATP_Olambda__382,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_kc(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_382
tff(fact_8563_ATP_Olambda__383,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_aex(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_383
tff(fact_8564_ATP_Olambda__384,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( pp(aa(real,bool,aa(fun(real,real),fun(real,bool),aTP_Lamp_yj(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_384
tff(fact_8565_ATP_Olambda__385,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aby(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_385
tff(fact_8566_ATP_Olambda__386,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_hq(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_386
tff(fact_8567_ATP_Olambda__387,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_ht(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_387
tff(fact_8568_ATP_Olambda__388,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_388
tff(fact_8569_ATP_Olambda__389,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_gz(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_389
tff(fact_8570_ATP_Olambda__390,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_gb(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_390
tff(fact_8571_ATP_Olambda__391,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_sa(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_391
tff(fact_8572_ATP_Olambda__392,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_sz(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_392
tff(fact_8573_ATP_Olambda__393,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_eg(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_ef(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_393
tff(fact_8574_ATP_Olambda__394,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_ee(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_ed(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_394
tff(fact_8575_ATP_Olambda__395,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_df(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_de(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_395
tff(fact_8576_ATP_Olambda__396,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_cq(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_cp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_396
tff(fact_8577_ATP_Olambda__397,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_cb(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_397
tff(fact_8578_ATP_Olambda__398,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_jn(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_398
tff(fact_8579_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aio(set(A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(A,real,real_V7696804695334737415tation(A,Uu,Uua),Uub)),Uub) ) ).

% ATP.lambda_399
tff(fact_8580_ATP_Olambda__400,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_ha(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_400
tff(fact_8581_ATP_Olambda__401,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_gc(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_gb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_401
tff(fact_8582_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_uu(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_bg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_402
tff(fact_8583_ATP_Olambda__403,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jm(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_jl(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_403
tff(fact_8584_ATP_Olambda__404,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jk(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_jj(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_404
tff(fact_8585_ATP_Olambda__405,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ji(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_jh(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_405
tff(fact_8586_ATP_Olambda__406,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jg(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_jf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_406
tff(fact_8587_ATP_Olambda__407,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_je(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_jd(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_407
tff(fact_8588_ATP_Olambda__408,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jc(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_jb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_408
tff(fact_8589_ATP_Olambda__409,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_sb(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = groups7121269368397514597t_prod(I6,B,aa(A,fun(I6,B),aTP_Lamp_sa(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uub),Uu) ) ).

% ATP.lambda_409
tff(fact_8590_ATP_Olambda__410,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qj(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_410
tff(fact_8591_ATP_Olambda__411,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_qk(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_411
tff(fact_8592_ATP_Olambda__412,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_fo(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_412
tff(fact_8593_ATP_Olambda__413,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_ko(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_cu(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_413
tff(fact_8594_ATP_Olambda__414,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_uo(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_414
tff(fact_8595_ATP_Olambda__415,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_uh(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_415
tff(fact_8596_ATP_Olambda__416,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_jo(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_416
tff(fact_8597_ATP_Olambda__417,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_st(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_417
tff(fact_8598_ATP_Olambda__418,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_ui(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_418
tff(fact_8599_ATP_Olambda__419,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qg(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_419
tff(fact_8600_ATP_Olambda__420,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fp(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_420
tff(fact_8601_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_lg(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_421
tff(fact_8602_ATP_Olambda__422,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_cj(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_422
tff(fact_8603_ATP_Olambda__423,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_lh(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_423
tff(fact_8604_ATP_Olambda__424,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_ma(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_424
tff(fact_8605_ATP_Olambda__425,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_lt(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_425
tff(fact_8606_ATP_Olambda__426,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_agr(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_426
tff(fact_8607_ATP_Olambda__427,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_ia(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_427
tff(fact_8608_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_jw(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_428
tff(fact_8609_ATP_Olambda__429,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ju(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_429
tff(fact_8610_ATP_Olambda__430,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_lf(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_430
tff(fact_8611_ATP_Olambda__431,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_qc(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_431
tff(fact_8612_ATP_Olambda__432,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_li(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_432
tff(fact_8613_ATP_Olambda__433,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_afg(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_433
tff(fact_8614_ATP_Olambda__434,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_mj(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_434
tff(fact_8615_ATP_Olambda__435,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aej(list(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),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_435
tff(fact_8616_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_il(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_436
tff(fact_8617_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_ahg(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_437
tff(fact_8618_ATP_Olambda__438,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(nat,fun(set(A),bool),aTP_Lamp_le(set(A),fun(nat,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu))
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_438
tff(fact_8619_ATP_Olambda__439,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lj(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_439
tff(fact_8620_ATP_Olambda__440,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_ci(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_440
tff(fact_8621_ATP_Olambda__441,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_cl(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_441
tff(fact_8622_ATP_Olambda__442,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_cm(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_442
tff(fact_8623_ATP_Olambda__443,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_da(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_443
tff(fact_8624_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ade(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_444
tff(fact_8625_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_kx(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_445
tff(fact_8626_ATP_Olambda__446,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_iq(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_446
tff(fact_8627_ATP_Olambda__447,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_kr(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_447
tff(fact_8628_ATP_Olambda__448,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ir(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_448
tff(fact_8629_ATP_Olambda__449,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_kt(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_449
tff(fact_8630_ATP_Olambda__450,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ku(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_450
tff(fact_8631_ATP_Olambda__451,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ks(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_451
tff(fact_8632_ATP_Olambda__452,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ael(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_452
tff(fact_8633_ATP_Olambda__453,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_aep(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_453
tff(fact_8634_ATP_Olambda__454,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_mr(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_454
tff(fact_8635_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,real),fun(A,bool),aTP_Lamp_abf(set(A),fun(fun(A,real),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_455
tff(fact_8636_ATP_Olambda__456,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_la(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_456
tff(fact_8637_ATP_Olambda__457,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_lr(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_457
tff(fact_8638_ATP_Olambda__458,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_kb(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_458
tff(fact_8639_ATP_Olambda__459,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_ab(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_459
tff(fact_8640_ATP_Olambda__460,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lv(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_460
tff(fact_8641_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lw(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_461
tff(fact_8642_ATP_Olambda__462,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_jx(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_462
tff(fact_8643_ATP_Olambda__463,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_mc(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_463
tff(fact_8644_ATP_Olambda__464,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_jz(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_464
tff(fact_8645_ATP_Olambda__465,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_md(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_465
tff(fact_8646_ATP_Olambda__466,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_lb(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))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_466
tff(fact_8647_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ah(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_467
tff(fact_8648_ATP_Olambda__468,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aek(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_468
tff(fact_8649_ATP_Olambda__469,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hp(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_469
tff(fact_8650_ATP_Olambda__470,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_abr(real,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu)) ) ).

% ATP.lambda_470
tff(fact_8651_ATP_Olambda__471,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_abp(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu)) ) ).

% ATP.lambda_471
tff(fact_8652_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_abn(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_472
tff(fact_8653_ATP_Olambda__473,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_xk(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_473
tff(fact_8654_ATP_Olambda__474,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_xu(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua)) ) ) ).

% ATP.lambda_474
tff(fact_8655_ATP_Olambda__475,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hs(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_475
tff(fact_8656_ATP_Olambda__476,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_476
tff(fact_8657_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_og(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_477
tff(fact_8658_ATP_Olambda__478,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_478
tff(fact_8659_ATP_Olambda__479,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_of(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_479
tff(fact_8660_ATP_Olambda__480,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_ib(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_480
tff(fact_8661_ATP_Olambda__481,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aeq(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_481
tff(fact_8662_ATP_Olambda__482,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_ep(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_482
tff(fact_8663_ATP_Olambda__483,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_aw(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_483
tff(fact_8664_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_kd(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_484
tff(fact_8665_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(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),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_485
tff(fact_8666_ATP_Olambda__486,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_bl(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_486
tff(fact_8667_ATP_Olambda__487,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_wm(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_487
tff(fact_8668_ATP_Olambda__488,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qh(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_488
tff(fact_8669_ATP_Olambda__489,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_bt(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_489
tff(fact_8670_ATP_Olambda__490,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dl(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_490
tff(fact_8671_ATP_Olambda__491,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_uw(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_491
tff(fact_8672_ATP_Olambda__492,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_cu(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_492
tff(fact_8673_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bg(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_493
tff(fact_8674_ATP_Olambda__494,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_bk(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_494
tff(fact_8675_ATP_Olambda__495,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_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_495
tff(fact_8676_ATP_Olambda__496,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_cy(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_496
tff(fact_8677_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dg(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_497
tff(fact_8678_ATP_Olambda__498,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_mi(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_498
tff(fact_8679_ATP_Olambda__499,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: fun(B,real),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,real),fun(B,bool),aTP_Lamp_yu(fun(B,real),fun(fun(B,real),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(B,real,Uu,Uub)),aa(B,real,Uua,Uub))) ) ).

% ATP.lambda_499
tff(fact_8680_ATP_Olambda__500,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_yq(fun(B,A),fun(fun(B,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)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_500
tff(fact_8681_ATP_Olambda__501,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_yr(fun(B,A),fun(fun(B,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)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_501
tff(fact_8682_ATP_Olambda__502,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_abt(fun(A,B),fun(fun(A,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)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_502
tff(fact_8683_ATP_Olambda__503,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_zc(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_503
tff(fact_8684_ATP_Olambda__504,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zu(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_504
tff(fact_8685_ATP_Olambda__505,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_ry(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_505
tff(fact_8686_ATP_Olambda__506,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_rl(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_506
tff(fact_8687_ATP_Olambda__507,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_zp(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_507
tff(fact_8688_ATP_Olambda__508,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_uc(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_508
tff(fact_8689_ATP_Olambda__509,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_gs(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_509
tff(fact_8690_ATP_Olambda__510,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_aao(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_510
tff(fact_8691_ATP_Olambda__511,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_aar(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_511
tff(fact_8692_ATP_Olambda__512,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_pg(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_512
tff(fact_8693_ATP_Olambda__513,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_uj(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_513
tff(fact_8694_ATP_Olambda__514,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_xy(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_514
tff(fact_8695_ATP_Olambda__515,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yk(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_515
tff(fact_8696_ATP_Olambda__516,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_afr(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_516
tff(fact_8697_ATP_Olambda__517,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ip(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_517
tff(fact_8698_ATP_Olambda__518,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aac(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_518
tff(fact_8699_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahw(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_519
tff(fact_8700_ATP_Olambda__520,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_io(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_520
tff(fact_8701_ATP_Olambda__521,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & 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_rb(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_521
tff(fact_8702_ATP_Olambda__522,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_aat(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_522
tff(fact_8703_ATP_Olambda__523,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & 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_aas(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_523
tff(fact_8704_ATP_Olambda__524,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_tv(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_524
tff(fact_8705_ATP_Olambda__525,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & 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_tw(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_525
tff(fact_8706_ATP_Olambda__526,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_ty(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_526
tff(fact_8707_ATP_Olambda__527,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_tz(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_527
tff(fact_8708_ATP_Olambda__528,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_zo(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_528
tff(fact_8709_ATP_Olambda__529,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tx(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_529
tff(fact_8710_ATP_Olambda__530,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_gt(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_530
tff(fact_8711_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aan(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8712_ATP_Olambda__532,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),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8713_ATP_Olambda__533,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8714_ATP_Olambda__534,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_xh(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_534
tff(fact_8715_ATP_Olambda__535,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_ya(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_535
tff(fact_8716_ATP_Olambda__536,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_fa(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_536
tff(fact_8717_ATP_Olambda__537,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vr(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_537
tff(fact_8718_ATP_Olambda__538,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aft(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_538
tff(fact_8719_ATP_Olambda__539,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_bb(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_539
tff(fact_8720_ATP_Olambda__540,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_ahy(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_540
tff(fact_8721_ATP_Olambda__541,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_aay(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_541
tff(fact_8722_ATP_Olambda__542,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_td(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_542
tff(fact_8723_ATP_Olambda__543,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_te(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_543
tff(fact_8724_ATP_Olambda__544,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_eu(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_544
tff(fact_8725_ATP_Olambda__545,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_ra(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_545
tff(fact_8726_ATP_Olambda__546,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_ph(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_546
tff(fact_8727_ATP_Olambda__547,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_ul(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_547
tff(fact_8728_ATP_Olambda__548,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_tf(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_548
tff(fact_8729_ATP_Olambda__549,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_agc(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_549
tff(fact_8730_ATP_Olambda__550,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_kk(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_550
tff(fact_8731_ATP_Olambda__551,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_tc(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_551
tff(fact_8732_ATP_Olambda__552,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_fg(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_552
tff(fact_8733_ATP_Olambda__553,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_el(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_553
tff(fact_8734_ATP_Olambda__554,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_afu(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_554
tff(fact_8735_ATP_Olambda__555,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_to(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_555
tff(fact_8736_ATP_Olambda__556,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_mh(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_556
tff(fact_8737_ATP_Olambda__557,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_qe(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_557
tff(fact_8738_ATP_Olambda__558,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_abc(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8739_ATP_Olambda__559,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_vi(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_559
tff(fact_8740_ATP_Olambda__560,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_se(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_560
tff(fact_8741_ATP_Olambda__561,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_xc(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8742_ATP_Olambda__562,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_ti(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_562
tff(fact_8743_ATP_Olambda__563,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_abg(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_563
tff(fact_8744_ATP_Olambda__564,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_vx(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_564
tff(fact_8745_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_565
tff(fact_8746_ATP_Olambda__566,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_adf(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_566
tff(fact_8747_ATP_Olambda__567,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_add(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_567
tff(fact_8748_ATP_Olambda__568,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_mq(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_568
tff(fact_8749_ATP_Olambda__569,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_yo(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_569
tff(fact_8750_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aez(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_570
tff(fact_8751_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_mn(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).

% ATP.lambda_571
tff(fact_8752_ATP_Olambda__572,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abl(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y5: A] :
            ( pp(aa(A,bool,Uua,Y5))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y5)) ) ) ) ).

% ATP.lambda_572
tff(fact_8753_ATP_Olambda__573,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_pu(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_573
tff(fact_8754_ATP_Olambda__574,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_ta(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_574
tff(fact_8755_ATP_Olambda__575,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_xf(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_575
tff(fact_8756_ATP_Olambda__576,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( pp(aa(nat,bool,aa(set(A),fun(nat,bool),aTP_Lamp_aad(fun(nat,set(A)),fun(set(A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_576
tff(fact_8757_ATP_Olambda__577,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jv(fun(nat,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,Uu,Uub)),Uua)) ) ).

% ATP.lambda_577
tff(fact_8758_ATP_Olambda__578,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ou(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_578
tff(fact_8759_ATP_Olambda__579,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ze(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_579
tff(fact_8760_ATP_Olambda__580,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_yw(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_580
tff(fact_8761_ATP_Olambda__581,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_zb(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_581
tff(fact_8762_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_yt(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_582
tff(fact_8763_ATP_Olambda__583,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_cw(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_583
tff(fact_8764_ATP_Olambda__584,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_ba(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_584
tff(fact_8765_ATP_Olambda__585,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_mu(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_585
tff(fact_8766_ATP_Olambda__586,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_ud(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_586
tff(fact_8767_ATP_Olambda__587,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_ev(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_587
tff(fact_8768_ATP_Olambda__588,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_pj(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_588
tff(fact_8769_ATP_Olambda__589,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_vc(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_589
tff(fact_8770_ATP_Olambda__590,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_yz(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_590
tff(fact_8771_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_8772_ATP_Olambda__592,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_ua(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_592
tff(fact_8773_ATP_Olambda__593,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_qw(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8774_ATP_Olambda__594,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_aal(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_594
tff(fact_8775_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aav(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_595
tff(fact_8776_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tr(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_596
tff(fact_8777_ATP_Olambda__597,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tt(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_597
tff(fact_8778_ATP_Olambda__598,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_es(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_598
tff(fact_8779_ATP_Olambda__599,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),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_599
tff(fact_8780_ATP_Olambda__600,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rr(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_600
tff(fact_8781_ATP_Olambda__601,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_bx(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_601
tff(fact_8782_ATP_Olambda__602,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),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_602
tff(fact_8783_ATP_Olambda__603,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_sr(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_603
tff(fact_8784_ATP_Olambda__604,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_8785_ATP_Olambda__605,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_605
tff(fact_8786_ATP_Olambda__606,axiom,
    ! [K9: $tType,L5: $tType,Uu: fun(K9,set(L5)),Uua: set(L5),Uub: K9] : aa(K9,set(L5),aa(set(L5),fun(K9,set(L5)),aTP_Lamp_op(fun(K9,set(L5)),fun(set(L5),fun(K9,set(L5))),Uu),Uua),Uub) = aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),minus_minus(set(L5)),aa(K9,set(L5),Uu,Uub)),Uua) ).

% ATP.lambda_606
tff(fact_8787_ATP_Olambda__607,axiom,
    ! [E4: $tType,F: $tType,Uu: fun(E4,set(F)),Uua: set(F),Uub: E4] : aa(E4,set(F),aa(set(F),fun(E4,set(F)),aTP_Lamp_ow(fun(E4,set(F)),fun(set(F),fun(E4,set(F))),Uu),Uua),Uub) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(E4,set(F),Uu,Uub)),Uua) ).

% ATP.lambda_607
tff(fact_8788_ATP_Olambda__608,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_tb(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_608
tff(fact_8789_ATP_Olambda__609,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_py(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_609
tff(fact_8790_ATP_Olambda__610,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_ru(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_610
tff(fact_8791_ATP_Olambda__611,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_pv(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_611
tff(fact_8792_ATP_Olambda__612,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_zq(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_612
tff(fact_8793_ATP_Olambda__613,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_ue(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_613
tff(fact_8794_ATP_Olambda__614,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xj(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_614
tff(fact_8795_ATP_Olambda__615,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(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_615
tff(fact_8796_ATP_Olambda__616,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_qd(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_616
tff(fact_8797_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yb(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_617
tff(fact_8798_ATP_Olambda__618,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_abz(fun(C,A),fun(int,fun(C,A)),Uu),Uua),Uub) = power_int(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_618
tff(fact_8799_ATP_Olambda__619,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_ace(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_619
tff(fact_8800_ATP_Olambda__620,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_acg(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_620
tff(fact_8801_ATP_Olambda__621,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_acd(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_621
tff(fact_8802_ATP_Olambda__622,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_ach(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_622
tff(fact_8803_ATP_Olambda__623,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_acf(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_623
tff(fact_8804_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ahu(fun(A,B),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,Uu,Uub)),Uua)) ) ).

% ATP.lambda_624
tff(fact_8805_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahr(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_625
tff(fact_8806_ATP_Olambda__626,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_kw(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_626
tff(fact_8807_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ik(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_627
tff(fact_8808_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bu(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_628
tff(fact_8809_ATP_Olambda__629,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aaf(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub))) ) ) ).

% ATP.lambda_629
tff(fact_8810_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mo(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_630
tff(fact_8811_ATP_Olambda__631,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_aab(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub)))) ) ) ).

% ATP.lambda_631
tff(fact_8812_ATP_Olambda__632,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_agy(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_632
tff(fact_8813_ATP_Olambda__633,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_zz(fun(A,B),fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua)) ) ) ).

% ATP.lambda_633
tff(fact_8814_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: fun(A,real)] :
          ( pp(aa(fun(A,real),bool,aa(A,fun(fun(A,real),bool),aTP_Lamp_aik(set(A),fun(A,fun(fun(A,real),bool)),Uu),Uua),Uub))
        <=> ( ! [V6: A] :
                ( ( aa(A,real,Uub,V6) != zero_zero(real) )
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),Uu)) )
            & pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Uub))))
            & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ahe(fun(A,real),fun(A,A),Uub)),collect(A,aTP_Lamp_ahf(fun(A,real),fun(A,bool),Uub))) = Uua ) ) ) ) ).

% ATP.lambda_634
tff(fact_8815_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aee(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite(A),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_635
tff(fact_8816_ATP_Olambda__636,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_ahm(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),vEBT_rec_VEBT(A,Uu,Uua,Uub)) ).

% ATP.lambda_636
tff(fact_8817_ATP_Olambda__637,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_ih(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_637
tff(fact_8818_ATP_Olambda__638,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_ii(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_638
tff(fact_8819_ATP_Olambda__639,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_ig(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_639
tff(fact_8820_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_acc(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_640
tff(fact_8821_ATP_Olambda__641,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(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(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_641
tff(fact_8822_ATP_Olambda__642,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gi(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_642
tff(fact_8823_ATP_Olambda__643,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahn(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_643
tff(fact_8824_ATP_Olambda__644,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_fn(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_644
tff(fact_8825_ATP_Olambda__645,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_wb(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_645
tff(fact_8826_ATP_Olambda__646,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_ex(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_646
tff(fact_8827_ATP_Olambda__647,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gj(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_647
tff(fact_8828_ATP_Olambda__648,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ahj(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),aa(set(A),set(B),image(A,B,Uu),Uua))) ) ) ).

% ATP.lambda_648
tff(fact_8829_ATP_Olambda__649,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_yy(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_649
tff(fact_8830_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_zd(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_650
tff(fact_8831_ATP_Olambda__651,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_za(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_651
tff(fact_8832_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_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),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_652
tff(fact_8833_ATP_Olambda__653,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_my(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_653
tff(fact_8834_ATP_Olambda__654,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_yx(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_654
tff(fact_8835_ATP_Olambda__655,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_afe(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_655
tff(fact_8836_ATP_Olambda__656,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_az(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_656
tff(fact_8837_ATP_Olambda__657,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_aam(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_657
tff(fact_8838_ATP_Olambda__658,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_et(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_658
tff(fact_8839_ATP_Olambda__659,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_by(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_659
tff(fact_8840_ATP_Olambda__660,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_vb(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_660
tff(fact_8841_ATP_Olambda__661,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_sq(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_661
tff(fact_8842_ATP_Olambda__662,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_662
tff(fact_8843_ATP_Olambda__663,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_ub(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_663
tff(fact_8844_ATP_Olambda__664,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_qx(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_664
tff(fact_8845_ATP_Olambda__665,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aau(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_665
tff(fact_8846_ATP_Olambda__666,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ts(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_666
tff(fact_8847_ATP_Olambda__667,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tu(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_667
tff(fact_8848_ATP_Olambda__668,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_xo(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_668
tff(fact_8849_ATP_Olambda__669,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_pl(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_669
tff(fact_8850_ATP_Olambda__670,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_oq(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_670
tff(fact_8851_ATP_Olambda__671,axiom,
    ! [G3: $tType,H4: $tType,Uu: set(G3),Uua: fun(H4,set(G3)),Uub: H4] : aa(H4,set(G3),aa(fun(H4,set(G3)),fun(H4,set(G3)),aTP_Lamp_or(set(G3),fun(fun(H4,set(G3)),fun(H4,set(G3))),Uu),Uua),Uub) = aa(set(G3),set(G3),aa(set(G3),fun(set(G3),set(G3)),minus_minus(set(G3)),Uu),aa(H4,set(G3),Uua,Uub)) ).

% ATP.lambda_671
tff(fact_8852_ATP_Olambda__672,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_wg(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_672
tff(fact_8853_ATP_Olambda__673,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_mv(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_673
tff(fact_8854_ATP_Olambda__674,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_abm(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_674
tff(fact_8855_ATP_Olambda__675,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aib(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtermap(B,A,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_675
tff(fact_8856_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_aaw(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_676
tff(fact_8857_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_ve(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_677
tff(fact_8858_ATP_Olambda__678,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_tq(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_678
tff(fact_8859_ATP_Olambda__679,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_abx(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_679
tff(fact_8860_ATP_Olambda__680,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aew(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_680
tff(fact_8861_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_he(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_8862_ATP_Olambda__682,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(C,A),Uua: real,Uub: C] :
          ( pp(aa(C,bool,aa(real,fun(C,bool),aTP_Lamp_yp(fun(C,A),fun(real,fun(C,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(A,aa(C,A,Uu,Uub)))) ) ) ).

% ATP.lambda_682
tff(fact_8863_ATP_Olambda__683,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_gw(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_683
tff(fact_8864_ATP_Olambda__684,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_ff(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_8865_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vo(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_685
tff(fact_8866_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),minus_minus(nat),Uub),Uua)) ) ).

% ATP.lambda_686
tff(fact_8867_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_sv(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_687
tff(fact_8868_ATP_Olambda__688,axiom,
    ! [Uu: fun(real,bool),Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_zg(fun(real,bool),fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua))) ) ).

% ATP.lambda_688
tff(fact_8869_ATP_Olambda__689,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_xn(fun(real,A),fun(real,fun(real,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_689
tff(fact_8870_ATP_Olambda__690,axiom,
    ! [Uu: fun(nat,real),Uua: nat,Uub: nat] : aa(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahv(fun(nat,real),fun(nat,fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_690
tff(fact_8871_ATP_Olambda__691,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bs(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_691
tff(fact_8872_ATP_Olambda__692,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_yv(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_692
tff(fact_8873_ATP_Olambda__693,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_sx(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_693
tff(fact_8874_ATP_Olambda__694,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_uq(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_694
tff(fact_8875_ATP_Olambda__695,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_ug(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_695
tff(fact_8876_ATP_Olambda__696,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_su(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_696
tff(fact_8877_ATP_Olambda__697,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_ca(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_697
tff(fact_8878_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rq(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_8879_ATP_Olambda__699,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,nat),Uub: nat] : aa(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_afq(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_699
tff(fact_8880_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_adn(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_700
tff(fact_8881_ATP_Olambda__701,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_oy(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_701
tff(fact_8882_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_nw(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_702
tff(fact_8883_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8884_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_rd(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_8885_ATP_Olambda__705,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_wu(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_705
tff(fact_8886_ATP_Olambda__706,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aaq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_706
tff(fact_8887_ATP_Olambda__707,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_pq(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_707
tff(fact_8888_ATP_Olambda__708,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afl(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_708
tff(fact_8889_ATP_Olambda__709,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_ady(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_709
tff(fact_8890_ATP_Olambda__710,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_it(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_710
tff(fact_8891_ATP_Olambda__711,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_adl(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_711
tff(fact_8892_ATP_Olambda__712,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_adm(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_712
tff(fact_8893_ATP_Olambda__713,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_km(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_713
tff(fact_8894_ATP_Olambda__714,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_kj(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_714
tff(fact_8895_ATP_Olambda__715,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_sy(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_715
tff(fact_8896_ATP_Olambda__716,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pp(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_716
tff(fact_8897_ATP_Olambda__717,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: fun(A,C),Uua: fun(C,D),Uub: A] : aa(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_xi(fun(A,C),fun(fun(C,D),fun(A,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(A,C,Uu,Uub)) ) ).

% ATP.lambda_717
tff(fact_8898_ATP_Olambda__718,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xg(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_718
tff(fact_8899_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_ahs(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).

% ATP.lambda_719
tff(fact_8900_ATP_Olambda__720,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_oj(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_720
tff(fact_8901_ATP_Olambda__721,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_uy(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_ux(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_721
tff(fact_8902_ATP_Olambda__722,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_sp(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_so(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_722
tff(fact_8903_ATP_Olambda__723,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aiq(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,real,real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_723
tff(fact_8904_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ain(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,real,real_V7696804695334737415tation(A,Uu,Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_724
tff(fact_8905_ATP_Olambda__725,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_xr(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_725
tff(fact_8906_ATP_Olambda__726,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_xq(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_726
tff(fact_8907_ATP_Olambda__727,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_xw(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_727
tff(fact_8908_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_bh(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_728
tff(fact_8909_ATP_Olambda__729,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_hc(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_729
tff(fact_8910_ATP_Olambda__730,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_wl(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_730
tff(fact_8911_ATP_Olambda__731,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_mw(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_731
tff(fact_8912_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_lo(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),collect(B,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ln(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_732
tff(fact_8913_ATP_Olambda__733,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [Uu: fun(A,real),Uua: fun(real,Aa),Uub: A] : aa(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_sm(fun(A,real),fun(fun(real,Aa),fun(A,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(Aa,aa(real,Aa,Uua,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_733
tff(fact_8914_ATP_Olambda__734,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_aih(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),aa(A,A,real_V8093663219630862766scaleR(A,K3),Uu))),real_Vector_span(A,Uua))) ) ) ).

% ATP.lambda_734
tff(fact_8915_ATP_Olambda__735,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_acn(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X5: B] :
          ( ( Uub = aa(B,A,Uu,X5) )
          & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),Uua)) ) ) ).

% ATP.lambda_735
tff(fact_8916_ATP_Olambda__736,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_acm(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X5: B] :
          ( ( Uub = aa(B,A,Uu,X5) )
          & pp(aa(B,bool,Uua,X5)) ) ) ).

% ATP.lambda_736
tff(fact_8917_ATP_Olambda__737,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aah(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),N4))
             => 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_or7035219750837199246ssThan(nat,Uub,N4)))),aa(nat,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_737
tff(fact_8918_ATP_Olambda__738,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_aag(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A6: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),A6))
             => ! [B7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B7))
                 => 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,A6,B7)))),aa(nat,real,Uua,A6))) ) ) ) ) ).

% ATP.lambda_738
tff(fact_8919_ATP_Olambda__739,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_acq(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N4: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N4),Uu),Uua) ) ).

% ATP.lambda_739
tff(fact_8920_ATP_Olambda__740,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_ed(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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_740
tff(fact_8921_ATP_Olambda__741,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_cp(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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_741
tff(fact_8922_ATP_Olambda__742,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_ef(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),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_742
tff(fact_8923_ATP_Olambda__743,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_adi(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),aa(set(A),set(B),image(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_743
tff(fact_8924_ATP_Olambda__744,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_ei(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_744
tff(fact_8925_ATP_Olambda__745,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_hg(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_745
tff(fact_8926_ATP_Olambda__746,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_xp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_746
tff(fact_8927_ATP_Olambda__747,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_hh(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_747
tff(fact_8928_ATP_Olambda__748,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_ej(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_748
tff(fact_8929_ATP_Olambda__749,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_kl(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_749
tff(fact_8930_ATP_Olambda__750,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_lz(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_750
tff(fact_8931_ATP_Olambda__751,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_ly(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_751
tff(fact_8932_ATP_Olambda__752,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_ls(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_752
tff(fact_8933_ATP_Olambda__753,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_aht(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_753
tff(fact_8934_ATP_Olambda__754,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_mt(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_754
tff(fact_8935_ATP_Olambda__755,axiom,
    ! [A: $tType,B: $tType,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_adb(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_755
tff(fact_8936_ATP_Olambda__756,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_mx(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_756
tff(fact_8937_ATP_Olambda__757,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_fx(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_fw(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_757
tff(fact_8938_ATP_Olambda__758,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_er(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_eq(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_758
tff(fact_8939_ATP_Olambda__759,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_dt(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_759
tff(fact_8940_ATP_Olambda__760,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_aip(set(A),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,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_760
tff(fact_8941_ATP_Olambda__761,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_dn(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_dm(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_761
tff(fact_8942_ATP_Olambda__762,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_di(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_dh(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_762
tff(fact_8943_ATP_Olambda__763,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_qp(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_763
tff(fact_8944_ATP_Olambda__764,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_qn(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_764
tff(fact_8945_ATP_Olambda__765,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_qm(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_765
tff(fact_8946_ATP_Olambda__766,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_ql(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_766
tff(fact_8947_ATP_Olambda__767,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_fj(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_767
tff(fact_8948_ATP_Olambda__768,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_nd(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_768
tff(fact_8949_ATP_Olambda__769,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_fw(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_769
tff(fact_8950_ATP_Olambda__770,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_wi(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_770
tff(fact_8951_ATP_Olambda__771,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_acy(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_771
tff(fact_8952_ATP_Olambda__772,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_zy(fun(A,B),fun(B,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_772
tff(fact_8953_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: real,Uuc: B] :
          ( pp(aa(B,bool,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_zk(fun(B,A),fun(A,fun(real,fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_773
tff(fact_8954_ATP_Olambda__774,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_dp(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_774
tff(fact_8955_ATP_Olambda__775,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_dc(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_775
tff(fact_8956_ATP_Olambda__776,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_dk(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_776
tff(fact_8957_ATP_Olambda__777,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_ds(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_777
tff(fact_8958_ATP_Olambda__778,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_fl(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_778
tff(fact_8959_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_nx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),aa(set(B),set(A),image(B,A,Uu),Uua)))
        & pp(aa(A,bool,Uub,Uuc)) ) ) ).

% ATP.lambda_779
tff(fact_8960_ATP_Olambda__780,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ln(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_780
tff(fact_8961_ATP_Olambda__781,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_lm(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_781
tff(fact_8962_ATP_Olambda__782,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,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_782
tff(fact_8963_ATP_Olambda__783,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_ob(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_783
tff(fact_8964_ATP_Olambda__784,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_ok(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_784
tff(fact_8965_ATP_Olambda__785,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahq(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uua))
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_785
tff(fact_8966_ATP_Olambda__786,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fm(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),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_786
tff(fact_8967_ATP_Olambda__787,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_db(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_787
tff(fact_8968_ATP_Olambda__788,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_jl(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_788
tff(fact_8969_ATP_Olambda__789,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_nj(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_789
tff(fact_8970_ATP_Olambda__790,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_jj(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_790
tff(fact_8971_ATP_Olambda__791,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_nh(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_791
tff(fact_8972_ATP_Olambda__792,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_nl(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_792
tff(fact_8973_ATP_Olambda__793,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_nn(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_793
tff(fact_8974_ATP_Olambda__794,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_agu(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_794
tff(fact_8975_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ny(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uua))
        & pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).

% ATP.lambda_795
tff(fact_8976_ATP_Olambda__796,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_ka(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_796
tff(fact_8977_ATP_Olambda__797,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_jy(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_797
tff(fact_8978_ATP_Olambda__798,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_wq(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_798
tff(fact_8979_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_bn(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_799
tff(fact_8980_ATP_Olambda__800,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_in(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_800
tff(fact_8981_ATP_Olambda__801,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_so(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_801
tff(fact_8982_ATP_Olambda__802,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_ux(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_802
tff(fact_8983_ATP_Olambda__803,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_sh(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_803
tff(fact_8984_ATP_Olambda__804,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_gl(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_804
tff(fact_8985_ATP_Olambda__805,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_zs(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_805
tff(fact_8986_ATP_Olambda__806,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_am(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_806
tff(fact_8987_ATP_Olambda__807,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_dm(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_807
tff(fact_8988_ATP_Olambda__808,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_de(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_808
tff(fact_8989_ATP_Olambda__809,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_dh(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_809
tff(fact_8990_ATP_Olambda__810,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aga(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_810
tff(fact_8991_ATP_Olambda__811,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_eq(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_811
tff(fact_8992_ATP_Olambda__812,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_aev(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_812
tff(fact_8993_ATP_Olambda__813,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_sl(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_813
tff(fact_8994_ATP_Olambda__814,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_sj(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_814
tff(fact_8995_ATP_Olambda__815,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rg(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_815
tff(fact_8996_ATP_Olambda__816,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_ri(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_816
tff(fact_8997_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_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,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_817
tff(fact_8998_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_qt(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_818
tff(fact_8999_ATP_Olambda__819,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_rt(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_819
tff(fact_9000_ATP_Olambda__820,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_qv(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_820
tff(fact_9001_ATP_Olambda__821,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_wr(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_821
tff(fact_9002_ATP_Olambda__822,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_wv(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_822
tff(fact_9003_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aed(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,finite_finite(B),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_823
tff(fact_9004_ATP_Olambda__824,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_zr(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_824
tff(fact_9005_ATP_Olambda__825,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_xa(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_wz(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_wz(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_wz(A,A)))))) ) ).

% ATP.lambda_825
tff(fact_9006_ATP_Olambda__826,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_wx(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_826
tff(fact_9007_ATP_Olambda__827,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_ww(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_827
tff(fact_9008_ATP_Olambda__828,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_ol(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_ok(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_828
tff(fact_9009_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_od(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,Uub,collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_ob(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_829
tff(fact_9010_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_oc(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uub),collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_ob(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_830
tff(fact_9011_ATP_Olambda__831,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_zj(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_bg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua))))) ) ) ).

% ATP.lambda_831
tff(fact_9012_ATP_Olambda__832,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_ie(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_832
tff(fact_9013_ATP_Olambda__833,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_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_833
tff(fact_9014_ATP_Olambda__834,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_gx(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_834
tff(fact_9015_ATP_Olambda__835,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_fz(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_835
tff(fact_9016_ATP_Olambda__836,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_jh(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_836
tff(fact_9017_ATP_Olambda__837,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_jf(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),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_837
tff(fact_9018_ATP_Olambda__838,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_ro(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_838
tff(fact_9019_ATP_Olambda__839,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_jb(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),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_839
tff(fact_9020_ATP_Olambda__840,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_jd(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_840
tff(fact_9021_ATP_Olambda__841,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_re(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_841
tff(fact_9022_ATP_Olambda__842,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_sd(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_sc(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_842
tff(fact_9023_ATP_Olambda__843,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_fv(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_fu(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_843
tff(fact_9024_ATP_Olambda__844,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_qo(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_qn(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_844
tff(fact_9025_ATP_Olambda__845,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_gm(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_gl(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_845
tff(fact_9026_ATP_Olambda__846,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_dw(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_846
tff(fact_9027_ATP_Olambda__847,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_dq(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_847
tff(fact_9028_ATP_Olambda__848,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_dr(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_848
tff(fact_9029_ATP_Olambda__849,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_fu(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_849
tff(fact_9030_ATP_Olambda__850,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(C,A),Uua: A,Uub: fun(C,B),Uuc: B,Uud: C] :
          ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_zf(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(C,B,Uub,Uud),Uuc)),real_V557655796197034286t_dist(A,aa(C,A,Uu,Uud),Uua))) ) ) ).

% ATP.lambda_850
tff(fact_9031_ATP_Olambda__851,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_rv(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_851
tff(fact_9032_ATP_Olambda__852,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_aci(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_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I2)),X_12))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I2)) ) )
        & ( ( Uuc = Uud )
         => ! [X5: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_12)) ) )
        & ( ( 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)
            & ! [X5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),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,X5)
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),X5))
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_852
tff(fact_9033_ATP_Olambda__853,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_aca(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_853
tff(fact_9034_ATP_Olambda__854,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
          ( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_agl(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A6: A,B7: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A6)),aa(C,B,Uua,B7)) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uub))
              & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B7),Uuc)) ) ) ) ).

% ATP.lambda_854
tff(fact_9035_ATP_Olambda__855,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_sc(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,aa(I6,fun(A,B),Uub,Uue),Uud)),groups7121269368397514597t_prod(I6,B,aa(A,fun(I6,B),aTP_Lamp_sa(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_855
tff(fact_9036_ATP_Olambda__856,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_rm(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_856
tff(fact_9037_ATP_Olambda__857,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_sf(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_857
tff(fact_9038_ATP_Olambda__858,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D,Uuc: fun(D,A),Uud: fun(D,A),Uue: D] : aa(D,A,aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_rc(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,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(D,A,Uu,Uub)),aa(D,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uua,Uue)),aa(D,A,Uuc,Uub))) ) ).

% ATP.lambda_858
tff(fact_9039_ATP_Olambda__859,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_rz(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_859
tff(fact_9040_ATP_Olambda__860,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ov(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_860
tff(fact_9041_ATP_Olambda__861,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_agk(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_861
tff(fact_9042_ATP_Olambda__862,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_agf(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_862
tff(fact_9043_ATP_Olambda__863,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_agb(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_863
tff(fact_9044_ATP_Olambda__864,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ags(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_864
tff(fact_9045_ATP_Olambda__865,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_age(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_865
tff(fact_9046_ATP_Olambda__866,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_agd(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_866
tff(fact_9047_ATP_Olambda__867,axiom,
    ! [Uu: rat,Uua: nat] : aa(nat,rat,aTP_Lamp_afs(rat,fun(nat,rat),Uu),Uua) = Uu ).

% ATP.lambda_867
tff(fact_9048_ATP_Olambda__868,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( zero(Aa)
        & topological_t2_space(Aa)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: Aa,Uua: A] : aa(A,Aa,aTP_Lamp_sw(Aa,fun(A,Aa),Uu),Uua) = Uu ) ).

% ATP.lambda_868
tff(fact_9049_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_qy(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_869
tff(fact_9050_ATP_Olambda__870,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_lq(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_870
tff(fact_9051_ATP_Olambda__871,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_nz(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_871
tff(fact_9052_ATP_Olambda__872,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_om(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_872
tff(fact_9053_ATP_Olambda__873,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_be(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_873
tff(fact_9054_ATP_Olambda__874,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pi(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_874
tff(fact_9055_ATP_Olambda__875,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: C] : aa(C,A,aTP_Lamp_lk(A,fun(C,A),Uu),Uua) = Uu ) ).

% ATP.lambda_875
tff(fact_9056_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ld(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_876
tff(fact_9057_ATP_Olambda__877,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mf(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_877
tff(fact_9058_ATP_Olambda__878,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_oa(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_878
tff(fact_9059_ATP_Olambda__879,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_eo(complex,complex),Uu) = Uu ).

% ATP.lambda_879
tff(fact_9060_ATP_Olambda__880,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_gh(nat,nat),Uu) = Uu ).

% ATP.lambda_880
tff(fact_9061_ATP_Olambda__881,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_hl(int,int),Uu) = Uu ).

% ATP.lambda_881
tff(fact_9062_ATP_Olambda__882,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_xe(C,C),Uu) = Uu ) ).

% ATP.lambda_882
tff(fact_9063_ATP_Olambda__883,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wz(A,A),Uu) = Uu ) ).

% ATP.lambda_883
tff(fact_9064_ATP_Olambda__884,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xb(A,A),Uu) = Uu ) ).

% ATP.lambda_884
tff(fact_9065_ATP_Olambda__885,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xd(A,A),Uu) = Uu ) ).

% ATP.lambda_885
tff(fact_9066_ATP_Olambda__886,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_nb(A,A),Uu) = Uu ) ).

% ATP.lambda_886
tff(fact_9067_ATP_Olambda__887,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ag(A,A),Uu) = Uu ) ).

% ATP.lambda_887
tff(fact_9068_ATP_Olambda__888,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_nt(A,A),Uu) = Uu ).

% ATP.lambda_888
tff(fact_9069_ATP_Olambda__889,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_agm(nat,rat),Uu) = zero_zero(rat) ).

% ATP.lambda_889
tff(fact_9070_ATP_Olambda__890,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ay(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_890
tff(fact_9071_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_au(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_891
tff(fact_9072_ATP_Olambda__892,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ek(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_892
tff(fact_9073_ATP_Olambda__893,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_lc(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_893
tff(fact_9074_ATP_Olambda__894,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_qz(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_894
tff(fact_9075_ATP_Olambda__895,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A] : aa(A,real,aTP_Lamp_ail(A,real),Uu) = zero_zero(real) ) ).

% ATP.lambda_895
tff(fact_9076_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_af(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_896
tff(fact_9077_ATP_Olambda__897,axiom,
    ! [Uu: nat] : aa(nat,rat,aTP_Lamp_agp(nat,rat),Uu) = one_one(rat) ).

% ATP.lambda_897
tff(fact_9078_ATP_Olambda__898,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_lp(A,real),Uu) = one_one(real) ).

% ATP.lambda_898
tff(fact_9079_ATP_Olambda__899,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_hv(real,bool),Uu))
    <=> $false ) ).

% ATP.lambda_899
tff(fact_9080_ATP_Olambda__900,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_iv(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_900
tff(fact_9081_ATP_Olambda__901,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_lx(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_901
tff(fact_9082_ATP_Olambda__902,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_iu(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_902
tff(fact_9083_ATP_Olambda__903,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_oz(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_903

% Type constructors (767)
tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_1,axiom,
    ! [A10: $tType] : bounded_lattice(filter(A10)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_2,axiom,
    bounded_lattice(bool) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
    ! [A10: $tType] : bounded_lattice(set(A10)) ).

tff(tcon_fun___Lattices_Obounded__lattice_4,axiom,
    ! [A10: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounded_lattice(fun(A10,A20)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => counta4013691401010221786attice(fun(A10,A20)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(fun(A10,A20)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( counta3822494911875563373attice(A20)
     => counta3822494911875563373attice(fun(A10,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(fun(A10,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A10: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4967611905675639751up_bot(fun(A10,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A10: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4346867609351753570nf_top(fun(A10,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(fun(A10,A20)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A10: $tType,A20: $tType] :
      ( boolea8198339166811842893lgebra(A20)
     => boolea8198339166811842893lgebra(fun(A10,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A10: $tType,A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(fun(A10,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A10: $tType,A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A10: $tType,A20: $tType] :
      ( order_top(A20)
     => order_top(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A10: $tType,A20: $tType] :
      ( order_bot(A20)
     => order_bot(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A10: $tType,A20: $tType] :
      ( preorder(A20)
     => preorder(fun(A10,A20)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A10: $tType,A20: $tType] :
      ( lattice(A20)
     => lattice(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A10: $tType,A20: $tType] :
      ( order(A20)
     => order(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A10: $tType,A20: $tType] :
      ( top(A20)
     => top(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A10: $tType,A20: $tType] :
      ( ord(A20)
     => ord(fun(A10,A20)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A10: $tType,A20: $tType] :
      ( bot(A20)
     => bot(fun(A10,A20)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A10: $tType,A20: $tType] :
      ( uminus(A20)
     => uminus(fun(A10,A20)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A10: $tType,A20: $tType] :
      ( minus(A20)
     => minus(fun(A10,A20)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_5,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___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_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_6,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_7,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_8,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_9,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_10,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_11,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_12,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_13,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_14,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_15,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_16,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_17,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_18,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_19,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_20,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_21,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_22,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_23,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_24,axiom,
    strict9044650504122735259up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
    ordere1170586879665033532d_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_25,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_26,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_27,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_28,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_29,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_30,axiom,
    ordere8940638589300402666id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
    canoni5634975068530333245id_add(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_31,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_32,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_33,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_34,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_42,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_43,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_44,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_45,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_46,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_47,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_48,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_49,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_50,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_51,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_52,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_53,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_54,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_55,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_56,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,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_61,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_77,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_78,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_79,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_80,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_81,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_82,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_83,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_84,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_85,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_86,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_87,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_88,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_89,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_90,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_91,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_92,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_93,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_94,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_95,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_96,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_97,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_98,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_99,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_100,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_101,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_102,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_103,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_104,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_105,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_106,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_107,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_108,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_109,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_110,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_111,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_112,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_113,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_114,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_115,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_116,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_117,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_118,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_119,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_120,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_121,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_122,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_123,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_124,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_125,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_126,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_127,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_128,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_129,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_130,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_131,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_132,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_133,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_134,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_135,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_136,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_137,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_138,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_139,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_140,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_141,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_142,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_143,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_144,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_145,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_146,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_147,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_148,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_149,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_150,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_151,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_152,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_153,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_154,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_155,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_156,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_157,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_158,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_159,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_160,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_161,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_162,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_163,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_164,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_165,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_166,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_167,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_168,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_169,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_170,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_171,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_172,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_173,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_174,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_175,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_176,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_177,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_178,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_179,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_180,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_181,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_182,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_183,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_184,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_185,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_186,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_187,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_188,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_189,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_190,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_191,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_192,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_193,axiom,
    ! [A10: $tType] : counta4013691401010221786attice(set(A10)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_194,axiom,
    ! [A10: $tType] : condit1219197933456340205attice(set(A10)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_195,axiom,
    ! [A10: $tType] : counta3822494911875563373attice(set(A10)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_196,axiom,
    ! [A10: $tType] : comple592849572758109894attice(set(A10)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_197,axiom,
    ! [A10: $tType] : bounde4967611905675639751up_bot(set(A10)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_198,axiom,
    ! [A10: $tType] : bounde4346867609351753570nf_top(set(A10)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_199,axiom,
    ! [A10: $tType] : comple6319245703460814977attice(set(A10)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_200,axiom,
    ! [A10: $tType] : boolea8198339166811842893lgebra(set(A10)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_201,axiom,
    ! [A10: $tType] : semilattice_sup(set(A10)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_202,axiom,
    ! [A10: $tType] : semilattice_inf(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_203,axiom,
    ! [A10: $tType] : order_top(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_204,axiom,
    ! [A10: $tType] : order_bot(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_205,axiom,
    ! [A10: $tType] : preorder(set(A10)) ).

tff(tcon_Set_Oset___Lattices_Olattice_206,axiom,
    ! [A10: $tType] : lattice(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oorder_207,axiom,
    ! [A10: $tType] : order(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Otop_208,axiom,
    ! [A10: $tType] : top(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oord_209,axiom,
    ! [A10: $tType] : ord(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Obot_210,axiom,
    ! [A10: $tType] : bot(set(A10)) ).

tff(tcon_Set_Oset___Groups_Ouminus_211,axiom,
    ! [A10: $tType] : uminus(set(A10)) ).

tff(tcon_Set_Oset___Groups_Ominus_212,axiom,
    ! [A10: $tType] : minus(set(A10)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_213,axiom,
    counta4013691401010221786attice(bool) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_214,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_215,axiom,
    counta3822494911875563373attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_216,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_217,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_218,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_219,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_220,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_221,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_222,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_223,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_224,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_225,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_226,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_227,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_228,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_229,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_230,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_231,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_232,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Otop_233,axiom,
    top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_234,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Orderings_Obot_235,axiom,
    bot(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_236,axiom,
    uminus(bool) ).

tff(tcon_HOL_Obool___Groups_Ominus_237,axiom,
    minus(bool) ).

tff(tcon_List_Olist___Nat_Osize_238,axiom,
    ! [A10: $tType] : size(list(A10)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_239,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_240,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_241,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_242,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_243,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_244,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_245,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_246,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_247,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_248,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_249,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_250,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_251,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_252,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_253,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_254,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_255,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_256,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_257,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_258,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_259,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_260,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_261,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_262,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_263,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_264,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_265,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_266,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_267,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_268,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_269,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_270,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_271,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_272,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_273,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_274,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_275,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_276,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_277,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_278,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_279,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_280,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_281,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_282,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_283,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_284,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_285,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_286,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_287,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_288,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_289,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_290,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_291,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_292,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_293,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_294,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_295,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_296,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_297,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_298,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_299,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_300,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_301,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_302,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_303,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_304,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_305,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_306,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_307,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_308,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_309,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_310,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_311,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_312,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_313,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_314,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_315,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_316,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_317,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_318,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_319,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_320,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_321,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_322,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_323,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_324,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_325,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_326,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_327,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_328,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_329,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_330,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_331,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_332,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_333,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_334,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_335,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_336,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_337,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_338,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_339,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_340,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_341,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_342,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_343,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_344,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_345,axiom,
    size(char) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_346,axiom,
    ! [A10: $tType] : condit1219197933456340205attice(filter(A10)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_347,axiom,
    ! [A10: $tType] : counta3822494911875563373attice(filter(A10)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_348,axiom,
    ! [A10: $tType] : bounde4967611905675639751up_bot(filter(A10)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_349,axiom,
    ! [A10: $tType] : bounde4346867609351753570nf_top(filter(A10)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_350,axiom,
    ! [A10: $tType] : comple6319245703460814977attice(filter(A10)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_351,axiom,
    ! [A10: $tType] : semilattice_sup(filter(A10)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_352,axiom,
    ! [A10: $tType] : semilattice_inf(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_353,axiom,
    ! [A10: $tType] : order_top(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_354,axiom,
    ! [A10: $tType] : order_bot(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_355,axiom,
    ! [A10: $tType] : preorder(filter(A10)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_356,axiom,
    ! [A10: $tType] : lattice(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_357,axiom,
    ! [A10: $tType] : order(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_358,axiom,
    ! [A10: $tType] : top(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_359,axiom,
    ! [A10: $tType] : ord(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_360,axiom,
    ! [A10: $tType] : bot(filter(A10)) ).

tff(tcon_Option_Ooption___Nat_Osize_361,axiom,
    ! [A10: $tType] : size(option(A10)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_362,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_363,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_364,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_365,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_366,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_367,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_368,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_369,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_370,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_371,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_372,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_373,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_374,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_375,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_376,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_377,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_378,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_379,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_380,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_381,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_382,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_383,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_384,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_385,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_386,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_387,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_388,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_389,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_390,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_391,axiom,
    real_V6936659425649961206t_norm(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_392,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_393,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_394,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_395,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_396,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_397,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_398,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_399,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_400,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_401,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_402,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_403,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_404,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_405,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_406,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_407,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_408,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_409,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_410,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_411,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_412,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_413,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_414,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_415,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_416,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_417,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_418,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_419,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_420,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_421,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_422,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_423,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_424,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_425,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_426,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_427,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_428,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_429,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_430,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_431,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_432,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_433,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_434,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_435,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_436,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_437,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_438,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_439,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_440,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_441,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_442,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_443,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_444,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_445,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_446,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_447,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_448,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_449,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_450,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_451,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_452,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_453,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_454,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_455,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_456,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_457,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_458,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_459,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_460,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_461,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_462,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_463,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_464,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_465,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_466,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_467,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_468,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_469,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_470,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_471,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_472,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_473,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_474,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_475,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_476,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_477,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_478,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_479,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_480,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_481,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_482,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_483,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_484,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_485,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_486,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_487,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_488,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_489,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_490,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_491,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_492,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_493,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_494,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_495,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_496,axiom,
    ! [A10: $tType,A20: $tType] :
      ( ( topolo4958980785337419405_space(A10)
        & topolo4958980785337419405_space(A20) )
     => topolo4958980785337419405_space(product_prod(A10,A20)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_497,axiom,
    ! [A10: $tType,A20: $tType] :
      ( ( topological_t2_space(A10)
        & topological_t2_space(A20) )
     => topological_t2_space(product_prod(A10,A20)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_498,axiom,
    ! [A10: $tType,A20: $tType] : size(product_prod(A10,A20)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_499,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_500,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_501,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_502,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_503,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_504,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_505,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_506,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_507,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_508,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_509,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_510,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_511,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_512,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_513,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_514,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_515,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_516,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_517,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_518,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_519,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_520,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_521,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_522,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_523,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_524,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_525,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_526,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_527,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_528,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_529,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_530,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_531,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_532,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_533,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_534,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_535,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_536,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_537,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_538,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_539,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_540,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_541,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_542,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_543,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_544,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_545,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_546,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_547,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_548,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_549,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_550,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_551,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_552,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_553,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_554,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_555,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_556,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_557,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_558,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_559,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_560,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_561,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_562,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_563,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_564,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_565,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_566,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_567,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_568,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_569,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_570,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_571,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_572,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_573,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_574,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_575,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_576,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_577,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_578,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_579,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_580,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_581,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_582,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_583,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_584,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_585,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_586,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_587,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_588,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_589,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_590,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_591,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_592,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_593,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_594,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_595,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_596,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_597,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_598,axiom,
    size(vEBT_VEBT) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y4: A] : if(A,fFalse,X,Y4) = Y4 ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y4: A] : if(A,fTrue,X,Y4) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(B,C),Q: fun(A,B),R: A] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,fun(B,C)),Q: B,R: A] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,fun(B,C)),Q: fun(A,B),R: A] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y4: A] :
      ( ( X != Y4 )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y4)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y4: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y4))
      | ( X = Y4 ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))))) ).

%------------------------------------------------------------------------------